A fuzzy MCDM approach for evaluating banking performance based on Balanced Scorecard

A fuzzy MCDM approach for evaluating banking performance based on

Balanced Scorecard

Hung-Yi Wu

a,*

, Gwo-Hshiung Tzeng

a,b

, Yi-Hsuan Chen

c

a

Department of Business and Entrepreneurial Management, Kainan University, No. 1, Kainan Road, Luchu, Taoyuan 338, Taiwan b_{Institute of Management of Technology, National Chiao-Tung University, 1001 Ta-Hsueh Road, Hsinchu 300, Taiwan} c_{Division of Global Purchasing, JI-EE Industry Co., Ltd., No. 498, Sec. 2, Bentian St., An Nan Dist., Tainan 709, Taiwan}

a r t i c l e

i n f o

Keywords: FMCDM

Balance Scorecard (BSC)

Fuzzy Analytic Hierarchy Process (FAHP) TOPSIS

VIKOR

a b s t r a c t

The paper proposed a Fuzzy Multiple Criteria Decision Making (FMCDM) approach for banking perfor-mance evaluation. Drawing on the four perspectives of a Balanced Scorecard (BSC), this research first summarized the evaluation indexes synthesized from the literature relating to banking performance. Then, for screening these indexes, 23 indexes fit for banking performance evaluation were selected through expert questionnaires. Furthermore, the relative weights of the chosen evaluation indexes were calculated by Fuzzy Analytic Hierarchy Process (FAHP). And the three MCDM analytical tools of SAW, TOPSIS, and VIKOR were respectively adopted to rank the banking performance and improve the gaps with three banks as an empirical example. The analysis results highlight the critical aspects of evaluation criteria as well as the gaps to improve banking performance for achieving aspired/desired level. It shows that the proposed FMCDM evaluation model of banking performance using the BSC framework can be a useful and effective assessment tool.

Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Financial liberalization and internationalization have been heavily advocated in Taiwan over the past decade, in response to increased global competition. Due to the government’s loosening control over the applications of establishing the medium-and-small business banks, the number of domestic headquarters and branches of financial institutions has increased from 6,127 to

6,365 between the years 2000 and 2005 (Central Bank of the

Republic of China, 2005). Financial institutions are densely distrib-uted in Taiwan. Moreover, the financial environment of Taiwan has undergone a drastic change since Taiwan entered the World Trade Organization (WTO). It is very important for Taiwan’s bank institu-tions to have a competitive advantage, because they are all quite homogeneous. Therefore, a fiercely competing financial market with relatively little profit, plus the new withdrawal mechanism regulations for low performance banks has resulted in a limited growth of banks in Taiwan. To outperform competing bank institu-tions, more emphasis on internal operational performance is re-quired. This means it is imperative to develop an effective way to conduct performance evaluations that can measure the overall

organizational performance and link it to the corporate goals. That is, a holistic evaluation model of banking performance is key to a bank’s survival.

Many different theories and methods of performance for con-ducting an evaluation have been applied in various organizations for many years. These approaches include ratio analysis, total pro-duction analysis, regression analysis, Delphi analysis, Balanced Scorecard, Analytic Hierarchical Process (AHP), Data Envelopment Analysis (DEA) and others. Each method has its own basic concept,

aim, advantages and disadvantages (Dessler, 2000). Which one is

chosen by management or decision makers for assessing perfor-mance depends on the status and type of the organization. How-ever, all the successful enterprises have some common features, including a specific vision, positive actions, and an effective perfor-mance evaluation. The Balanced Scorecard (BSC) is an extensive and thorough performance evaluation tool to adequately plan and control an organization so it can attain its goals (Davis & Alb-right, 2004; Lawrie & Cobbold, 2004; Pinero, 2002). The BSC breaks through the traditional limitations of finance, examining an organi-zation’s performance from the four main perspectives of finance, customer, internal business process, and learning and growth (Kaplan & Norton, 1992). It emphasizes both the aspects of the financial and non-financial, long-term and short-term strategies, and emphasizes internal and external business measures. Several studies have been conducted incorporating the four perspectives of the BSC in performance appraisal. To achieve the best possible 0957-4174/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved.

doi:10.1016/j.eswa.2009.01.005

* Corresponding author. Tel.: +886 33412500x6103; fax: +886 33412176. E-mail addresses:hywu@mail.knu.edu.tw,whydec66@yahoo.com.tw(H.-Y. Wu),

ghtzeng@mail.knu.edu.tw(G.-H. Tzeng),j02255@ji-ee.com.tw(Y.-H. Chen).

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Expert Systems with Applications

result from a more effective performance, it is crucial to improve the banking relationship by matching the needs of the clients to the delivery process of client services (Nist, 1996). Therefore, the BSC is also utilized as a framework to develop evaluation indicators for banking performance (Davis & Albright, 2004; Kim & Davidson, 2004; Kuo & Chen, 2010).

SinceBellman and Zadeh (1970)developed the theory of

deci-sion behavior in a fuzzy environment, various relevant models were developed, and have been applied to different fields such as control engineering, artificial intelligence, management science, and Multiple Criteria Decision Making (MCDM) among others. The concept of combining the fuzzy theory and MCDM is referred to as fuzzy MCDM (FMCDM). Several practicable applications of utilizing FMCDM in criteria evaluation and alternatives selection are demonstrated in previous studies (Bayazita & Karpak, 2007; Chen, Lin, & Huang, 2006; Chiou & Tzeng, 2002; Chiou, Tzeng, & Cheng, 2005; Chiu, Chen, Shyu, & Tzeng, 2006; Hsieh, Lu, & Tzeng, 2004; Lee, Chen, & Chang, 2008; Pepiot, Cheikhrouhou, Furbringer, & Glardon, 2008; Wang & Chang, 2007; Wu & Lee, 2007). Primarily, the MCDM problems are first classified into distinct aspects and different alternatives/strategies and the criteria are defined based on various points of view from stakeholders. Then, a finite set of alternatives/strategies can be evaluated in terms of multi-criteria. Choosing a suitable method to measure the criteria can help the evaluators and analysts to process the cases to be evaluated and determine the best alternative. Like most cases of evaluation, a number of criteria have to be considered for performance apprai-sal. Consequently, banking performance evaluation can be re-garded as a MCDM problem. In addition, the multiple criteria used in the BSC are more objective and comprehensive than a sin-gle one. In this research, a FMCDM approach based on the four per-spectives of the BSC was proposed to establish a performance evaluation model for bank institutions. The aims of this research are as follows: (1) screen performance indexes to fit the banks for constructing a hierarchical framework of performance evalua-tion; (2) use FAHP (Fuzzy Analytic Hierarchy Process) to find the fuzzy weights of the indexes by subjective perception; (3) apply SAW (Simple Average Weight), TOPSIS (Technique for Order Prefer-ence by Similarity to Ideal Solution method), and VIKOR (VlseKri-terijumska Optimizacija I Kompromisno Resenje) to rank the performance and improve the gaps of three banks in the example; and (4) provide suggestions based on the research results for per-formance evaluation and serve as a reference for future research in this field.

The remainder of this paper is organized as follows. The con-cepts of performance evaluation and BSC are introduced and

re-viewed in Section 2. In Section 3, the performance evaluation

framework and the analytical methods used in FMCDM for

evalu-ating the banking performance are proposed. Section 4provides

an empirical example for banking performance, including the hier-archical framework of BSC performance evaluation indexes and the result analyses and discussion to illustrate the proposed

perfor-mance evaluation model. Section5concludes the paper.

2. Performance evaluation and Balanced Scorecard

This section briefly reviews the underlying concepts adopted by this research, such as the definitions of performance evalua-tion, performance evaluation index, and Balanced Scorecard (BSC).

2.1. Definitions of performance evaluation

The definitions of performance and evaluation are as follows. Performance is referred to as one kind of measurement of the goals

of an enterprise, while evaluation is referred to as the goal that an enterprise can effectively obtain during a specific period (Lebas, 1995). Evans, Ashworth, Chellew, Davidson, and Towers (1996) stated that performance evaluation is an important activity of management control, used to investigate whether resources are allocated efficiently; it is applied for the purpose of operational control to achieve a goal adjustment in the short-term and for strategy management and planning in the long run. As indicated by Rue and Byars (2005), performance evaluation tells us how employees define their own work, and it establishes a

decision-making and communication process for improvement. Kaplan

and Norton (1992)described performance evaluation as a way to review the achievements of organizations of both their financial and non-financial objectives.

There is abundant literature on performance evaluation demon-strating various topics and successful examples relating to

perfor-mance management (McNamara & Mong, 2005). The traditional

performance rankings of banks is based on simple and consistent factors such as financial returns, returns on asset (ROA) and returns on earning (ROE). Nevertheless, performance rankings conducted in this way may not precisely illustrate institutions that embrace strategies for sustaining top performance (Hanley & Suter, 1997). Non-financial criteria such as customer satisfaction, community and employee relations can be vital to a bank’s winning strategy, because using only ROA or ROE for performance ranking may not necessarily determine which institution offers the highest returns to the investors, nor does it accurately prove which one is most profitable.

Evaluations of the performance of a bank can be diverse ( Kosmi-dou, Pasiouras, Doumpos, & Zopounidis, 2006). Several previous studies on bank performance measurement examined economies of scale and scope employing traditional statistical methods such as correlation analysis (Arshadi & Lawrence, 1987), translog cost function (Gilligann, Smirlock, & Marshall, 1984; Molyneux, Altun-bas, & Gardener, 1996; Murray & White, 1983), loglinear models, or tools like Data Envelopment Analysis (DEA), etc. ( Athanassopo-ulos & Giokas, 2000; Drake, 2001; Giokas, 1991).

2.2. Performance evaluation index

Performance measurement can be defined as a system by which a company monitors its daily operations and evaluates whether the company is attaining its objectives. To fully utilize the function of performance measurement, it is suggested to set up a series of indexes which properly reflect the performance of a company. These indicators can be quantifiable, or unquantifiable. For in-stance, an index such as lead time is viewed as a quantifiable (or financial) measure, whereas the degree of customer satisfaction is unquantifiable (or non-financial) measures. Managers often have difficulty in delineating strategies and selecting proper measures while implementing the BSC system.

In the early stage of implementing the BSC, it is important to collect as many ideas as possible concerning performance mea-surement by interviewing business managers and discussing their

business vision, mission, and strategies. Meyer and Markiewicz

(1997)grouped the measures relating to the critical success factors of banking performance into eights categories: (1) profitability, (2) efficiency and productivity, (3) human resource management, (4) risk management, (5) sales effectiveness, (6) service quality, (7)

capital management, and (8) competitive positioning. Collier

(1995)employed structural equation models to analyze the pro-cess performance of banks using criteria such as propro-cess quality er-rors, employee turnover rate, labor productivity, on-time delivery,

and unit cost. The multidimensional indexes used byArshadi and

Lawrence (1987) include profitability, pricing of bank services, and loan market share.

The majority of past studies have focused on customers and how they choose the bank that will offer them general bank ser-vices. According to the related literature, the selection criteria which customers use to evaluate and choose between banks, in-clude price, speed, access, customer service, location, image and reputation, modern facilities, interest rates, opening hours, incen-tive offered, product range, and service charge policy and so on (Anderson, Cox, & Fulcher, 1976; Boyd, Leonard, & White, 1994; Chia & Hoon, 2000; Devlin, 2002; Devlin & Gerrard, 2005; Elliot, Shatto, & Singer, 1996; Martenson, 1985). The more recent

re-search ofDevlin and Gerrard (2005)made an attempt to address

the relative importance of various choice criteria in the selection of a banking institution by applying a quantitative methodology of statistical analysis. They provided an analysis of customer choice criteria and multiple banking and made an itemized comparison of the relative importance of choice criteria which impact on the choice for main and secondary banking institutions.

2.3. Balanced Scorecard

The concept of Balanced Scorecard (BSC) was proposed by David Norton, the CEO of Nolan Norton Institute, and Robert Kaplan, a professor at Harvard University (Kaplan & Norton, 1992). The BSC measures organizational performance from four perspectives, including financial, customer, internal business process, and learn-ing and growth, in relation to the four functions of accountlearn-ing and finance, marketing, value chain, and human resource. These mea-sures, both financial and non-financial, from all four perspectives serve as the common language to help align the top management and employees toward with the organization’s vision. The BSC pro-vides managers with the instrumentation tools they need to navi-gate towards future competitive success (Kaplan & Norton, 1992, 1996a, 1996b). The essential tenet of the BSC is that standard financial measures must be balanced with non-financial measures (Norton, Contrada, & LoFrumento, 1997).

There has been generally accepted in practice that since the introduction of the BSC by Kaplan and Norton a combination of financial and non-financial measures in a performance measure-ment system is favorable for both profit and non-profit organiza-tions (Ballou, Heitger, & Tabor, 2003; Sinclair & Zairi, 2001). Banks can save both time and money if they recognize which mea-sures are most suitable for their needs. Non-financial meamea-sures such as intangibles like customer relationships may account for more than half of the total assets of a company. An important prin-ciple of the BSC is to achieve success on key non-financial mea-sures before actualizing success on key financial meamea-sures. When considered in non-financial measures to other measures, these metrics can lead organizations to administer performance

effec-tively and forecast their future profitability (Anonymous, 2006;

Mouritsen, Thorsgaard, & Bukh, 2005).

The BSC is a popular tool that is applied by many businesses to assess their performance in diverse aspects of their organization. It provides insights into corporate performance not only for manag-ers seeking ways to improve performance, but also for investors wanting to gauge the organizations’ ongoing health. For banks the benefits of using BSC are numerous: (1) can be used as a frame-work to assess and develop a bank’s strategy; (2) can be used to de-velop strategic objectives and performance measures to transform a bank’s strategy into action; (3) it provides a way to measure and monitor the performance of key performance drivers that may lead to the successful execution of a bank’s strategy; and (4) it is an effective tool to ensure that a bank continuously improves its sys-tem and process (Frigo, Pustorino, & Krull, 2000).Davis and Alb-right (2004) presented an empirical analysis that explores the effect of the BSC on a banking institution’s financial performance. Kim and Davidson (2004)used the BSC framework to assess the

business performance of information technology (IT) expenditures in the banking industry using the t-test and regression models.Kuo and Chen (2010)applied the four perspectives of the BSC to con-struct key performance appraisal indicators for the mobility of

the service industries through the fuzzy Delphi method. Leung,

Lam, and Cao (2006) proposed a tailor-made performance mea-surement model using the analytic hierarchy process and the ana-lytic network process for implementing the BSC.

A large amount of research related to the financial industry em-ployed the BSC to evaluate performance and has benefited from its use (Ashton, 1998; Davis & Albright, 2004). Nevertheless, most of these studies focused on how to set up an effective mechanism to select evaluation criteria rather than on calculating their relative weight. Therefore, this research aims at developing an evaluation model for banking performance not only to investigate the relative importance among the selection criteria, but also to examine the critical gaps for achieving aspired/desired level.

3. Performance evaluation framework and analytical methods The analytical structure of this research is illustrated inFig. 1. A performance analysis is conducted based on the selected evalua-tion criteria. First the FAHP approach is employed to calculate the relative weights of the performance evaluation indexes. Then, according to these weights the three MCDM analytical tools of SAW, TOPSIS, and VIKOR are used to rank and improve the banking performance and determine the best practice. The concepts of the fuzzy set theory and details of the analytical methods are ex-plained in the following subsections.

3.1. Fuzzy set theory

Expressions such as ‘‘not very clear”, ‘‘probably so”, and ‘‘very likely”, are used often in daily life, and more or less represent some degree of uncertainty of human thought. The fuzzy set theory pro-posed byZadeh (1965), an important concept applied in the scien-tific environment, has been available to other fields as well. Consequently, the fuzzy theory has become a useful tool for auto-mating human activities with uncertainty-based information. Therefore, this research incorporates the fuzzy theory into the per-formance measurement by objectifying the evaluators’ subjective judgments.

Evaluation Index Selection for Banking Performance of the BSC

Analysis of Relative Importance Weights of Performance Evaluation

Index

Ranking of the Banks

FAHP

SAW/ TOPSIS/

VIKOR

Performance Analysis

3.1.1. Fuzzy number

In the classical set theory, the truth value of a statement can be given by the membership function as

l

A(x)

l

AðxÞ ¼

1 if x 2 A; 0 if x R A: 

ð1Þ

Fuzzy numbers are a fuzzy subset of real numbers, and they represent the expansion of the idea of a confidence interval.

According to the definition byDubois and Prade (1978), the fuzzy

number eA is of a fuzzy set, and its membership function is

l

_e

AðxÞ : R ! ½0; 1ð05

l

eAðxÞ51; x 2 XÞ, where x represents the

crite-rion and is described by the following characteristics: (1)

l

_e

AðxÞ is

a continuous mapping from R (real line) to the closed interval [0, 1]; (2)

l

_e

AðxÞ is of a convex fuzzy subset; (3)

l

eAðxÞ is the

normal-ization of a fuzzy subset, which means that there exists a number x0such that

l

_e

Aðx0Þ ¼ 1. For instance, the triangular fuzzy number

(TFN), eA ¼ ðl; m; uÞ, can be defined as Eq.(2)and the TFN member-ship function is shown inFig. 2:

l

_e AðxÞ ¼ ðx  lÞ=ðm  lÞ if l 6 x 6 m; ðu  xÞ=ðu  mÞ if m 6 x 6 u; 0 otherwise: 8 > < > : ð2Þ

Based on the characteristics of TFN and the extension defini-tions proposed byZadeh (1975), given any two positive triangular fuzzy numbers, eA1¼ ðl1;m1;u1Þ and eA2¼ ðl2;m2;u2Þ; and a positive

real number r, some algebraic operations of the triangular fuzzy numbers eA1and eA2can be expressed as follows:

Addition of two TFNs :

eA1 eA2¼ ðl1þ l2;m1þ m2;u1þ u2Þ: ð3Þ Multiplication of two TFNs :

eA1 eA2¼ ðl1l2;m1m2;u1u2Þ: ð4Þ Multiplication of any real number r and a TFN :

r  eA1¼ ðrl1;rm1;ru1Þ for r > 0 and li>0; mi>0; ui>0 ð5Þ Subtraction of two TFNsH: eA1

H

eA2¼ ðl1 u2;m1 m2;u1 l2Þ for li>0; mi>0; ui>0: ð6Þ Division of two TFNs £: A1£A2¼ ðl1=u2;m1=m2;u1=l2Þ: ð7Þ Reciprocal of a TFN: eA1 1 ¼ ð1=u1;1=m1;1=l1Þ for li>0; mi>0; ui>0: ð8Þ 3.1.2. Linguistic variable

Linguistic variables are variables whose values are words or sentences in a natural or artificial language. In other words, they are variables with lingual expression as their values (Hsieh et al.,

2004; Zadeh, 1975). The possible values for these variables could be: ‘‘very dissatisfied”, ‘‘not satisfied”, ‘‘fair”, ‘‘satisfied”, and ‘‘very satisfied”. The evaluators are asked to conduct their judgments, and each linguistic variable can be indicated by a triangular fuzzy number (TFN) within the scale range of 0–100. An example of membership functions of five levels of linguistic variables is shown inFig. 3. For instance, the linguistic variable ‘‘Satisfied” can be rep-resented as (60, 80, 100). Besides, each evaluator can personally de-fine his/her subjective range of linguistic variables. The use of linguistic variables is applied widely. In this paper, linguistic vari-ables expressed by TFN are adopted to stand for evaluators’ subjec-tive measures to determine the degrees of importance among evaluation criteria and also assess the performance value of alternatives.

3.2. Fuzzy analytic hierarchy process

The Analytic Hierarchy Process (AHP) was devised by Saaty

(1980, 1994). It is a useful approach to solve complex decision problems. It prioritizes the relative importance of a list of criteria (critical factors and sub-factors) through pairwise comparisons amongst the factors by relevant experts using a nine-point scale. Buckley (1985)incorporated the fuzzy theory into the AHP, called the Fuzzy Analytic Hierarchy Process (FAHP). It generalizes the cal-culation of the consistent ratio (CR) into a fuzzy matrix. The proce-dure of FAHP for determining the evaluation weights are explained as follows:

Step 1: Construct fuzzy pairwise comparison matrices. Through expert questionnaires, each expert is asked to assign

lin-guistic terms by TFN (as shown inTable 1andFig. 4) to

the pairwise comparisons among all criteria in the dimen-sions of a hierarchy system. The result of the comparisons is constructed as fuzzy pairwise comparison matrices ðeAÞ as shown in Eq.(9).

Step 2: Examine the consistency of the fuzzy pairwise comparison matrices. According to the research ofBuckley (1985), it proves that if A = [aij] is a positive reciprocal matrix then

e

A ¼ ½~aij is a fuzzy positive reciprocal matrix. That is, if

the result of the comparisons of A = [aij] is consistent, then

it can imply that the result of the comparisons of eA ¼ ½~aij

is also consistent. Therefore, this research employs this method to validate the questionnaire

eA ¼ 1 ~a12    ~a1n ~ a21 1    ~a2n .. . .. . . . . .. . ~ an1 ~an2    1 2 6 6 6 6 4 3 7 7 7 7 5¼ 1 _a~_{12   } ~_a1n 1=~a12 1    ~a2n .. . .. . . . . .. . 1=~a1n 1=~a2n    1 2 6 6 6 6 4 3 7 7 7 7 5: ð9Þ

Step 3: Compute the fuzzy geometric mean for each criterion. The geometric technique is used to calculate the geometric mean ð~riÞ of the fuzzy comparison values of criterion i to

each criterion, as shown in Eq.(10), where ~ainis a fuzzy

value of the pair-wise comparison of criterion i to criterion n (Buckley, 1985) ( ) Ax µ l m u 1 0 x ~

Fig. 2. Membership function of the triangular fuzzy number.

Satisfied Very

dissatisfied satisfiedNot Fair

Very satisfied 20 40 60 80 100 1 0 x 0.5 A x)µ~ ₍

~ri¼ ~½ai1     ~ain1=n: ð10Þ Step 4: Compute the fuzzy weights by normalization. The fuzzy weight of the ith criterion ð ~wiÞ, can be derived as Eq.

(11), where ~wiis denoted as ~wi¼ ðLwi;Mwi;UwiÞ by a TFN and Lwi, Mwi, and Uwi represent the lower, middle and upper values of the fuzzy weight of the ith criterion

~

wi¼ ~ri ð~r1 ~r2     ~rnÞ1: ð11Þ

3.3. The synthetic value of fuzzy judgment

SinceBellman and Zadeh (1970)proposed the decision-making

methods in fuzzy environments, an increasing number of related models have been applied in various fields, including control engi-neering, expert system, artificial intelligence, management science, operations research, and MCDM. The above decision-making prob-lems were solved by applying the fuzzy set theory. This approach is called the fuzzy MCDM (FMCDM). Its main application is focused on criteria evaluation or project selection. The FMCDM method can assist decision makers in selecting the best alternative, or rank-ing the order of projects.

Due to the differences in the subjective judgments among the experts for each evaluation criterion, the overall valuation of the fuzzy judgment is employed to synthesize the various experts’ opinions in order to achieve a reasonable and objective evaluation. The calculation steps to obtain the synthetic value are:

Step 1: Performance evaluation of the alternatives. As shown in Fig. 3, ‘‘very dissatisfied”, ‘‘not satisfied”, ‘‘fair”, ‘‘satisfied”, and ‘‘very satisfied” are the five linguistic variables used to measure the performance of the alternatives against the evaluation criteria. Each linguistic variable can be

pre-sented by a TFN with a range of 0–100. Assume that eEk

ij

denotes the fuzzy valuation of performance given by the evaluator k towards alternative i under criterion j as Eq. (12)shows, then: eEk ij¼ LE k ij;ME k ij;UE k ij   : ð12Þ

In this research, eEijrepresents the average fuzzy judgment

values integrated by m evaluators as

eEij¼ ð1=mÞ  eE1ij eE 2 ij     eE m ij   : ð13Þ

According toBuckley (1985), the three end points of eEijcan

be computed as LEij¼ Xm k¼1 LEkij !, m; MEij¼ Xm k¼1 MEkij !, m; UEij¼ Xm k¼1 UKk ij !, m: ð14Þ

Step 2: Fuzzy synthetic judgment. According to the fuzzy weight, ~

wj, of each criterion calculated by FAHP, the criteria vector

ð ~wÞ is derived as Eq. (15). And, the fuzzy performance matrix ðeEÞ, as presented in Eq.(16), of all the alternatives can be acquired from the fuzzy performance value of each alternative under n criteria.

~

w ¼ ð ~w1; . . . ; ~wj; . . . ; ~wnÞt; ð15Þ

eE ¼ ½~eij: ð16Þ

Then, the final fuzzy synthetic decision can be deduced from the cri-teria weight vector ð ~wÞ and the fuzzy performance matrix ðeEÞ; and then the derived result, the final fuzzy synthetic decision matrix ð eRÞ, is calculated by eR ¼ eE () ~w, where the sign, ,, indicates the computation of the fuzzy number, consisting of both fuzzy addi-tion and fuzzy multiplicaaddi-tion. Considering that the computaaddi-tion of fuzzy multiplication is rather complicated, the approximate multi-plied result of the fuzzy multiplication is used here. For instance, the approximate fuzzy number ðeRiÞ of the fuzzy synthetic decision

of the alternative i is denoted as Eq.(17), where LRi, MRi, and URiare

the lower, middle, and upper synthetic performance values of alter-native i, respectively, and the calculations of each are illustrated as Eq.(18) eRi¼ ðLRi;MRi;URiÞ; ð17Þ where LRi¼ Xn j¼1 Lwj LEij; MRi¼ Xn j¼1 Mwj MEij; URi¼ Xn j¼1 Uwj UEij: ð18Þ

Next, the procedure of defuzzification (Hsieh et al., 2004;

Opricovic & Tzeng, 2003) locates the Best Nonfuzzy Performance value (BNP). Methods used in such defuzzified fuzzy ranking gen-erally include the mean of maximal (MOM), center of area (COA), and

a

-cut. Utilizing the COA method to find out the BNP is a simple and practical without the need to bring in the preferences of any evaluators. Therefore it is used in this study. The BNP value of the fuzzy number eRican be found by

BNPi¼ ½ðURi LRiÞ þ ðMRi LRiÞ=3 þ LRi

8

i: ð19Þ The ranking of the alternatives then proceeds based on the value of the derived BNP for each of the alternatives.

3.4. TOPSIS method

TOPSIS was developed byHwang and Yoon (1981), based on the

concept that the chosen/improved alternatives should be the shortest distance from the positive ideal solution (PIS) and the Table 1

Membership function of the linguistic scale.

Fuzzy number Linguistic scales TFN ð~aijÞ Reciprocal of a TFN ð~aijÞ

~

9 Absolutely important (7, 9, 9) (1/9, 1/9, 1/7)

~

7 Very strongly important (5, 7, 9) (1/9, 1/7, 1/5)

~ 5 Essentially important (3, 5, 7) (1/7, 1/5, 1/3) ~ 3 Weakly important (1, 3, 5) (1/5, 1/3, 1) ~ 1 Equally important (1, 1, 3) (1/3, 1, 1) ~

2; ~4; ~6; ~8 Intermediate value between two adjacent judgments Source:Mon et al. (1994) and Hsieh et al. (2004).

x Equally

important importantWeakly Essentially important Very strongly important Absolutely important

1 3 5 7 9

1

( )

Ax

µ~

farthest from the negative-ideal solution (NIS) for solving a MCDM problem. Thus, the best alternative should not only be the shortest distance away from the positive ideal solution (aspired/desired le-vel), but also should be the largest distance away from the negative ideal solution (tolerable level). In short, the ideal solution is com-posed of all the criteria with the best values attainable (aspired/de-sired levels), whereas the negative ideal solution is made up of all the criteria with the worst values attainable (tolerable level). The general step-by-step procedure using the TOPSIS is briefly listed as follows.

Step 1: Establish the original performance matrix. The structure of

the performance matrix (X) is shown as Eq.(20), where

Aidenotes the alternative i, i = 1, 2, . . . , m; Cjis the jth

crite-rion, j = 1, 2, . . . , n. Therefore, xijrepresents the performance

value of alterative i in criterion j

ð20Þ

Step 2: Calculate the normalized performance matrix. The purpose of normalizing the performance (including: the larger is better and the smaller is better) is to remove the units of matrix entries by converting the performance values to a range between 0 and 1. The normalized value (rij) is

calcu-lated as rij¼ jxij xjj jx j  xjj ; i ¼ 1; 2; . . . ; m; j ¼ 1; 2; . . . ; n: ð21Þ

Step 3: Compute the weighted normalized performance matrix. Con-sidering that there is a difference in the importance of the criteria, the normalized performance matrix has to be weighted as illustrated in Eq.(22), where wjis the weight

of the criterion j, and

v

ijis the weighted normalized

perfor-mance matrix. The summation of wjis equal to 1

v

ij¼ wj rij: ð22Þ

Step 4: Determine the ideal (aspired/desired) and negative-ideal (tol-erable/worst) solutions. The ideal and negative ideal

solu-tions (A* _{and A}_{) are elaborated as Eqs. (23) and (24)}

respectively, where Cbis associated with the benefit

crite-ria, and Ccis associated with the cost criteria A

¼ fðmax

i

v

ijjj 2 CbÞ; ðmini

v

ijjj 2 CcÞg ð23Þ or setting the aspired/desired level

ðx 1; . . . ;x  j; . . . ;x  nÞ ¼

v

 jjj ¼ 1; 2; . . . ; n n o or A¼ fw1; . . . ;wj; . . . ;wng; A¼ fðmin i

v

ijjj 2 CbÞ; ðmaxi

v

ijjj 2 CcÞg ð24Þ or setting the tolerable level

ð0; . . . ; 0; . . . ; 0Þ ¼ f

v



jjj ¼ 1; 2; . . . ; ng or A 

¼ ð0; . . . ; 0; . . . ; 0Þ

Step 5: Calculate the separation measures. The distance can be cal-culated by the n-dimensional Euclidean distance. The sep-arations of each alternative from the ideal solution ðd

iÞ

and the negative-ideal solution ðd_iÞ are defined as Eqs. (25) and (26)respectively. di ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn j¼1 ð

v

ij

v

jÞ 2 v u u t

8

i; ð25Þ di ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Xn j¼1 ð

v

ij

v

jÞ 2 v u u t

8

i: ð26Þ

Step 6: Calculate the relative closeness (similarity) to the ideal solu-tion and rank the preference order. The relative closeness of alternative Aiwith respect to A*can be expressed as

Eq.(27), where the index value of RC

i is between 0 and 1. RC i ¼ di d iþ d  i ¼ 1  d  i d iþ d  i ; ð27Þ where RC

i shows that the larger the index value, the better the

per-formance of the alternatives and

di diþ d  i    i ¼ 1; 2; . . . ; m   ð28Þ

Eq.(28)denotes a relative indicator of the synthetic gap in alterna-tive i caused by j criterion and j = 1, 2, . . . , n. The synthetic gap is the main issue in this problem. How can we improve/reduce the gaps to reach zero so as to achieve the aspired/desired level in each crite-rion? The TOPSIS method used to provide the information for improving the gaps in each criterion cannot be used for ranking purpose (seeOpricovic & Tzeng, 2004, pp. 450–456 and Fig. 2). Therefore, the authors propose the VIKOR method for ranking and improving the alternatives of this problem. This method is intro-duced in the Section3.5.

3.5. VIKOR

The Multi-criteria Optimization and Compromise Solution (called VIKOR) is a suitable tool to evaluate each alternative for each criterion function (Opricovic & Tzeng, 2002, 2003, 2004, 2007; Tzeng, Lin, & Opricovic, 2005). The concept of VIKOR is based on the compromise programming of MCDM by comparing the measure of ‘‘closeness” to the ‘‘ideal” alternative. The multi-criteria

measure for compromise ranking is developed from the Lp-metric

that is used as an aggregating function in compromise

program-ming (Yu, 1973; Zeleny, 1982). The compromise ranking algorithm

of VIKOR consists of the following steps:

Step 1: Determine the be st (aspired/desired levels) and worst values (tolerable/worst levels). Assuming that jth criterion repre-sents a benefit, then the best values for setting all the

cri-teria functions (aspired/desired levels) are fx

jjj ¼ 1;

2; . . . ; ng and the worst values (tolerable/worst levels) are fx

jjj ¼ 1; 2; . . . ; ng, respectively.

Step 2: Compute the gaps fSiji ¼ 1; 2; . . . ; mg and fRiji ¼ 1; 2; . . . ; mg

from the Lp-metric referring to Eq.(29)by normalization.

The relationships are presented in Eqs.(30) and (31)

dpi ¼ Xn j¼1 wj jx j  xijj jx j xjj !p ( )1=p ; i ¼ 1; 2; . . . ; m; ð29Þ Si¼ dp¼1i ¼ Xn j¼1 wj jx j  xijj jx j  xjj ! ; i ¼ 1; 2; . . . ; m; ð30Þ Ri¼ dp¼1i ¼ max j wj jx j  xijj jx j  xjj jj ¼ 1; 2; . . . ; n ( ) ; i ¼ 1; 2; . . . ; m; ð31Þ

where Si, Ri2 [0, 1] and 0 denotes the best (i.e., achieving

Step 3: Compute the gaps fQiji ¼ 1; 2; . . . ; mg for ranking. The

rela-tion is defined as Eq.(32), where S _{¼ min}

i Si (the best S *

can be set equal zero), S ¼ max

i Si (the worst S

_{can be}

set to equal one); R _{¼ min}

i Ri(the best R

*_{can be set to}

equal zero), R ¼ max

i Ri(the worst R

_{can be set to equal}

one), and

v

2 [0, 1] is introduced as the weight of the strat-egy of the ‘‘the majority of the criteria” (or ‘‘maximum group utility”), usually

v

= 0.5. In this research, the value of

v

is set to equal 0, 0.5 and 1 for sensitive analysis.

Qi¼

v

ðSi SÞ ðS SÞ  þ ð1 

v

Þ ðRi R  Þ ðR RÞ  ; i ¼ 1; 2; . . . ; m; ð32Þ

Step 4: Rank and improve the alternatives, sort by the values S, R, and Q, in decreasing order and reduce the gaps in the criteria. The results are three ranking lists, with the best alternatives having the lowest value.

Step 5: Propose a compromise solution. For a given criteria weight,

the alternatives (a0_{), are the best ranked by measure Q}

(minimum) if the following two conditions are satisfied: C1. ‘‘Acceptable advantage”: Q(a00_{)  Q(a}0_{) P DQ, where a}00 _is

the alternative with second position in the ranking list by Q; DQ = 1/(J  1); J is the number of alternatives.

C2. ‘‘Acceptable stability in decision making”: Alternative a0

must also be the best ranked by S or/and R. This compro-mise solution is stable within a decision making process, which could be: ‘‘voting by majority rule” (when

v

> 0.5 is needed), or ‘‘by consensus”v 0.5, or ‘‘with veto” (

v

< 0.5). Here,

v

is the weight of decision making strategy ‘‘majority of criteria” (or ‘‘the maximum group utility”). If one of the conditions is not satisfied, then a set of compromise solutions is proposed, consisting of:

Alternatives a0_{and a}00 _{if only condition C2 is not satisfied, or}

Alternativesa0_{, a}00_{, . . . , a}(M)_{if condition C1 is not satisfied; and a}(M)

is determined by the relation Q(a00_{)  Q(a}0_{) < DQ for maximum M}

(the positions of these alternatives are ‘‘in closeness”).

The compromise solution obtained by VIKOR can be accepted by the decision makers because it provides a maximum ‘‘group util-ity” of the ‘‘majorutil-ity” (with measure S, representing ‘‘concor-dance”), and a minimum individual regret of an ‘‘opponent” (with measure R, representing ‘‘discordance”). The compromise solutions can be the basis for negotiations, by involving the criteria weights of the decision makers’ preference.

4. An empirical example for banking performance

The four perspectives of BSC were taken as the framework for establishing performance evaluation indexes in this research. Based on this framework, the FAHP was used to obtain the fuzzy weights of the indexes. The three MCDM analytical tools, SAW, TOPSIS, and VIKOR were respectively applied to evaluate the bank-ing performance based on the weight of each index, and to improve the gaps with three banks as an empirical example. The hierarchi-cal framework of the BSC performance evaluation criteria, and the results, analyses and discussions of the empirical example are illustrated in the following section.

4.1. Hierarchical framework of the BSC performance evaluation criteria

From the four BSC perspectives, and based on a review of the lit-erature, 55 evaluation indexes related to banking performance were summarized. Then, expert questionnaires were used for screening the indexes fit for the banking performance evaluation. Twenty-three evaluation indexes were selected by the committee of experts, comprised of twelve professionals from practice and the academia. The descriptions of the criteria for the selection eval-uation of a bank’s performance are listed in theappendix. The hier-archical framework of the BSC performance evaluation criteria (i.e. four dimensions and 23 indexes) for banking is shown inFig. 5. The 23 evaluation indexes are grouped into the four BSC dimensions, ‘‘F: Finance (F1–F6)”, ‘‘C: Customer (C1–C6)”, ‘‘P: Internal Process (P1–P6)”, and ‘‘L: Learning and Growth (L1–L5)”.

BSC performance e v aluation F: Finance C: Customer P: Internal Process

L: Learning and Growth

L1 Responses of customer service L2 Professional training L3 Employee stability L4 Employee satisfaction L5 Organization competence P1 No. of new service items

P3 Customer complaints P2 Transaction efficiency P4 Rationalized forms & processes

P6 Management performance P5 Sales performance s e x e d n I s n o i s n e m i D l a o G F1 Sales F3 Return on assets F2 Debt ratio F4 Earnings per share

F6 Return on inveatment F5 Net profit margin

C1 Customer satisfaction

C3 Market share rate C2 Profit per customer C4 Customer retention rate

C6 Profit per customer C5 Customer increasing rate

4.2. Weights of the evaluation criteria

Based on the hierarchical framework of the BSC performance evaluation indexes, the FAHP questionnaire using FTN was distrib-uted among the experts for soliciting their professional opinions. The relative importance (fuzzy weight) of each performance index analyzed by FAHP is listed inTable 2using Eqs.(10) and (11). The result shows that the critical order of the four BSC dimensions for banking performance evaluation is ‘‘C: Customer (0.4101)”, ‘‘F: Fi-nance (0.3271)”, ‘‘P: Internal process (0.1314)”, and ‘‘L: Learning and growth (0.1314)”. The top five important evaluation indexes are ‘‘C1: Customer satisfaction (0.1237)”, ‘‘F3: Return on assets (0.0812)”, ‘‘F4: Earnings per share (0.0784)”, ‘‘C4: Customer reten-tion rate (0.0741), ” and ‘‘C2: Profit per customer (0.0715)”. The least important evaluation index is ‘‘L1: Responses of customer ser-vice (0.0109)”.

4.3. Ranking of the banking performance

Three banks (e.g. C Bank, S Bank, and U Bank) were taken as an illustrative example and were evaluated by the experts based on the selected evaluation criteria. Since there are differences of sub-jective judgments among the way experts view each evaluation criterion, the overall evaluation of the fuzzy judgment was em-ployed to synthesize the opinions of the various experts in order to achieve a reasonable and objective evaluation. In this research, the five linguistic variables, ‘‘very dissatisfied”, ‘‘not satisfied”, ‘‘fair”, ‘‘satisfied”, and ‘‘very satisfied” were used to measure the banking performance with respect to the evaluation criteria. As shown inFig. 3, each linguistic variable is presented by a TFN with a range of 0–100. The average fuzzy judgment values of each crite-rion of the three banks, integrated by various experts through Eqs. (12)–(14), are summarized inTable 3. Then, the final fuzzy syn-thetic judgment of the three banks is deduced from the fuzzy cri-teria weights (Table 2) and the fuzzy judgment values (Table 3)

by Eqs.(15)–(18).Table 4presents the final fuzzy synthetic judg-ment of the three banks based on the evaluation criteria. Conse-quently, based on the fuzzy weights of the evaluation criteria calculated by FAHP, the three MCDM analytical tools, SAW, TOPSIS, and VIKOR were respectively adopted to rank the banking perfor-mance. First, the performance ranking order of the three banks using SAW is C Bank (113.97)  U Bank (113.65)  S Bank (107.99) as shown inTable 4.

The TOPSIS method was then applied to evaluate the banks’

per-formance. Referring toTable 3, the BNP values computed by Eq.

Table 2

Fuzzy weights of BSC performance evaluation index by FAHP.

Criteria (Dimension and index) Local weights Overall weights BNPa

STD_BNPb Rank (F) Finance (0.1293, 0.3325, 0.7592) 0.4070 0.3271 2 (F1) Sales (0.0830, 0.2020, 0.4371) (0.0107, 0.0672, 0.3318) 0.1366 0.0604 6 (F2) Debt ratio (0.0305, 0.0696, 0.2405) (0.0039, 0.0231, 0.1823) 0.0699 0.0309 14 (F3) Return on assets (0.0830, 0.2282, 0.6103) (0.0107, 0.0759, 0.4633) 0.1833 0.0812 2

(F4) Earnings per share (0.0807, 0.2348, 0.5824) (0.0104, 0.0781, 0.4422) 0.1769 0.0784 3

(F5) Net profit margin (0.0531, 0.1166, 0.3057) (0.0069, 0.0388, 0.2321) 0.0926 0.0410 10

(F6) Return on Investment (0.0580, 0.1489, 0.3997) (0.0075, 0.0495, 0.3035) 0.1202 0.0532 7

(C) Customer (0.1775, 0.4348, 0.9184) 0.5102 0.4101 1

(C1) Customer satisfaction (0.1104, 0.3179, 0.7404) (0.0197, 0.1382, 0.6800) 0.2793 0.1237 1

(C2) Profit per on-line customer (0.0894, 0.1960, 0.4171) (0.0159, 0.0852, 0.3831) 0.1614 0.0715 5

(C3) Market share rate (0.0609, 0.1370, 0.3117) (0.0108, 0.0596, 0.2863) 0.1189 0.0527 8

(C4) Customer retention rate (0.0703, 0.1792, 0.4476) (0.0125, 0.0779, 0.4111) 0.1672 0.0741 4 (C5) Customers increasing rate (0.0441, 0.0837, 0.2252) (0.0078, 0.0364, 0.2068) 0.0837 0.0371 13

(C6) Profit per customer (0.0450, 0.0861, 0.2394) (0.0080, 0.0374, 0.2199) 0.0884 0.0392 11

(P) Internal process (0.0643, 0.1164, 0.3098) 0.1635 0.1314 3

(P1) No. of new service items (0.1085, 0.3340, 0.8097) (0.0070, 0.0389, 0.2508) 0.0989 0.0438 9 (P2) Transaction efficiency (0.0878, 0.1936, 0.4171) (0.0056, 0.0225, 0.1292) 0.0525 0.0232 19

(P3) Customer complaints (0.0597, 0.1274, 0.2851) (0.0038, 0.0148, 0.0883) 0.0357 0.0158 20

(P4) Rationalized forms and processes (0.0691, 0.1771, 0.4476) (0.0044, 0.0206, 0.1387) 0.0546 0.0242 18

(P5) Sales performance (0.0433, 0.0828, 0.2252) (0.0028, 0.0096, 0.0698) 0.0274 0.0121 22

(P6) Management performance (0.0442, 0.0851, 0.2394) (0.0028, 0.0099, 0.0742) 0.0290 0.0128 21

(L) Learning and growth (0.0643, 0.1164, 0.3098) 0.1635 0.1314 3

(L1) Responses of customer service (0.0381, 0.0742, 0.2016) (0.0025, 0.0087, 0.0625) 0.0245 0.0109 23 (L2) Professional training (0.1289, 0.3234, 0.6899) (0.0083, 0.0376, 0.2137) 0.0866 0.0383 12

(L3) Employee stability (0.0967, 0.2014, 0.4392) (0.0062, 0.0234, 0.1361) 0.0552 0.0245 17

(L4) Employee satisfaction (0.0898, 0.1915, 0.4466) (0.0058, 0.0223, 0.1384) 0.0555 0.0246 16 (L5) Organization competence (0.0898, 0.2095, 0.4781) (0.0058, 0.0244, 0.1481) 0.0594 0.0263 15 a

BNP (Best non-fuzzy performance) = [(U  L) + (M  L)]/3 + L. b

STD_BNP: standardized BNP.

Table 3

Average fuzzy judgment values of each evaluation criterion by various experts.

Indexes C Bank S Bank U Bank

F1 (80.00, 100.00, 100.00) (73.33, 91.67, 96.67) (65.00, 81.25, 92.50) F2 (50.00, 66.67, 83.33) (60.00, 75.00, 90.00) (52.50, 68.75, 85.00) F3 (30.00, 50.00, 70.00) (30.00, 50.00, 70.00) (50.00, 66.67, 83.33) F4 (23.33, 41.67, 60.00) (40.00, 58.33, 76.67) (60.00, 75.00, 90.00) F5 (45.00, 62.50, 80.00) (52.50, 68.75, 85.00) (57.50, 75.00, 87.50) F6 (40.00, 56.25, 72.50) (45.00, 62.50, 80.00) (73.30, 91.70, 96.70) C1 (50.00, 66.70, 83.30) (37.50, 56.30, 75.00) (40.00, 58.30, 76.70) C2 (56.70, 75.00, 86.70) (23.30, 41.70, 60.00) (20.00, 37.50, 55.00) C3 (73.30, 91.70, 96.70) (32.50, 50.00, 67.50) (60.00, 75.00, 90.00) C4 (30.00, 50.00, 70.00) (40.00, 58.30, 76.70) (23.30, 41.70, 60.00) C5 (57.50, 75.00, 87.50) (60.00, 66.70, 83.30) (50.00, 66.70, 83.30) C6 (60.00, 75.00, 90.00) (45.00, 62.50, 80.00) (73.30, 91.70, 96.70) P1 (52.50, 68.75, 85.00) (45.00, 62.50, 80.00) (33.33, 50.00, 66.67) P2 (47.50, 62.50, 77.50) (40.00, 58.33, 76.67) (52.50, 68.75, 85.00) P3 (40.00, 58.33, 76.67) (60.00, 75.00, 90.00) (65.00, 81.25, 92.50) P4 (60.00, 75.00, 90.00) (50.00, 66.67, 83.33) (60.00, 75.00, 90.00) P5 (66.67, 83.33, 93.33) (66. 67, 83.33, 96.67) (50.00, 66.67, 83.33) P6 (50.00, 66.67, 83.33) (52.50, 68.75, 85.00) (66.67, 83.33, 93.33) L1 (40.00, 58.33, 76.67) (40.00, 58.33, 76.67) (73.33, 91.67, 96.67) L2 (66.67, 83.33, 93.33) (57.50, 75.00, 87.50) (60.00, 75.00, 90.00) L3 (23.33, 41.67, 60.00) (23.33, 41.67, 60.00) (40.00, 58.33, 76.67) L4 (45.00, 62.50, 80.00) (45.00, 62.50, 80.00) (56.67, 75.00, 86.67) L5 (50.00, 66.67, 90.00) (50.00, 66.67, 83.33) (50.00, 56.67, 83.33)

(19)of the average fuzzy judgment values of the three banks

inte-grated by various experts was given by Eq.(20)as shown inTable

5. The normalized performance matrix was obtained as

summa-rized in Table 6 by Eq. (21). According to the fuzzy weights of

the BSC performance evaluation indexes by FAHP as shown in

Ta-ble 2, the weighted normalized performance matrices calculated

by Eq. (22) and both positive ideal and negative ideal solutions

for the BSC evaluation criteria set by Eqs.(23) and (24)are shown inTable 7.Table 8lists the separations of the ideal solution ðdiÞ

and the negative-ideal solution ðd

iÞ for the three banks by Eqs.

(25) and (26). The relative closeness ðRC

iÞ to the ideal solution

and preference evaluation result derived from Eqs.(27) and (28)

by TOPSIS are presented inTable 9. The relative closeness ðRCiÞ

val-ues for the three banks are C Bank (RC*_{= 0.5579), U Bank}

(RC*_{= 0.4704), and S Bank (RC}*_{= 0.3521), respectively. This implies}

that C Bank has the smallest gap for achieving the aspired/desired level among the three banks, whereas S Bank has the largest gap.

Similarly, VIKOR was used to rank the banking performance of the three banks based on the fuzzy weights of the BSC performance

evaluation indexes by FAHP as shown inTable 2.Table 10shows

the performance matrix given by Eq.(20)with the best value x

j

(aspired/desired levels) and the worst value x

j (tolerable/worst

levels). The values of Siand Ri computed by Eqs. (29)–(31) are

shown inTable 11, while the computed value Qi(with

v

= 0, 0.5,

1) by Eq.(32)and the preference order ranking are given inTable 12. The performance ranking order of the three banks by VIKOR is C Bank (Qi= 0.0000)  U Bank (Qi= 0.3909)  S Bank (Qi= 1.0000).

Finally, the final values and preference order ranking by these three MCDM methods, SAW, TOPSIS, and VIKOR are summarized inTable 13.

It indicates that all the ranking results are identical. However, the final values of the three banks calculated by SAW and TOPISIS are extremely close to each other. In this case, the VIKOR method is found to be a better method of assessment to clearly discriminate the banking performance.

Table 4

Fuzzy synthetic performance values of the evaluation criteria by SAW.

Criteria C Bank S Bank U Bank

F (0.29, 6.81, 112.29) (0.32, 7.30, 120.58) (0.39, 8.44, 132.30) F1 (0.86, 6.71, 33.10) (0.78, 6.15, 32.00) (0.70, 5.45, 30.62) F2 (0.20, 1.53, 15.20) (0.23, 1.73, 16.42) (0.20, 1.58, 15.50) F3 (0.32, 3.79, 32.44) (0.32, 3.79, 32.44) (0.54, 5.05, 38.62) F4 (0.24, 3.25, 26.54) (0.42, 4.55, 33.91) (0.62, 5.85, 39.81) F5 (0.31, 2.42, 18.58) (0.36, 2.66, 19.74) (0.40, 2.90, 20.32) F6 (0.30, 2.78, 22.01) (0.34, 3.09, 24.29) (0.55, 4.53, 29.36) C (0.71, 13.26, 169.21) (0.49, 10.27, 146.66) (0.54, 10.80, 148.95) C1 (0.99, 9.22, 56.64) (0.74, 7.78, 51.00) (0.70, 5.45, 30.62) C2 (0.90, 6.39, 33.21) (0.37, 3.55, 22.99) (0.20, 1.58, 15.50) C3 (0.79, 5.47, 27.69) (0.35, 2.98, 19.33) (0.54, 5.05, 38.62) C4 (0.38, 3.90, 28.78) (0.50, 4.54, 31.53) (0.62, 5.85, 39.81) C5 (0.45, 2.73, 18.10) (0.47, 2.43, 17.23) (0.40, 2.90, 20.32) C6 (0.48, 2.81, 19.79) (0.36, 2.34, 17.59) (0.55, 4.53, 29.36) P (0.09, 0.93, 19.60) (0.08, 0.90, 19.37) (0.09, 0.09, 18.93) P1 (0.37, 2.67, 21.32) (0.32, 2.43, 20.06) (0.23, 1.95, 16.72) P2 (0.27, 1.41, 10.01) (0.22, 1.31, 9.91) (0.29, 1.55, 10.98) P3 (0.15, 0.86, 6.77) (0.23, 1.11, 7.95) (0.25, 1.20, 8.17) P4 (0.26, 1.55, 12.48) (0.22, 1.37, 11.56) (0.26, 1.55, 12.48) P5 (0.19, 0.80, 6.51) (0.19, 0.80, 6.75) (0.14, 0.64, 5.82) P6 (0.14, 0.66, 6.18) (0.15, 0.68, 6.31) (0.19, 0.82, 6.93) L (0.09, 0.89, 17.75) (0.08, 0.85, 17.06) (0.10, 0.94, 18.60) L1 (0.10, 0.51, 4.79) (0.10, 0.51, 4.79) (0.18, 0.80, 6.04) L2 (0.55, 3.13, 19.94) (0.48, 2.82, 18.70) (0.50, 2.82, 19.23) L3 (0.14, 0.98, 8.17) (0.14, 0.98, 8.17) (0.25, 1.36, 10.43) L4 (0.26, 1.39, 11.07) (0.26, 1.39, 11.07) (0.33, 1.67, 12.00) L5 (0.29, 1.63, 13.33) (0.29, 1.63, 12.34) (0.29, 1.38, 12.34) Synthetic performance (1.17, 21.89, 318.86) (0.98, 19.32, 303.68) (1.11, 21.06, 113.65) BNPa 113.97 107.99 113.65 Rankingb 1 3 2 a

BNP (Best non-fuzzy performance) = [(U  L) + (M  L)]/3 + L. b

Ranking: Rank by SAW.

Table 5

The performance matrix [xij]mnof three banks by various experts.

Indexes C Bank S Bank U Bank

F1 93.33 87.22 79.58 F2 66.67 75.00 68.75 F3 50.00 50.00 66.67 F4 41.67 58.33 75.00 F5 62.50 68.75 73.33 F6 56.25 62.50 87.23 C1 66.67 56.27 58.33 C2 72.80 41.67 37.50 C3 87.23 50.00 75.00 C4 50.00 58.33 41.67 C5 73.33 70.00 66.67 C6 75.00 62.50 87.23 P1 68.75 62.50 50.00 P2 62.50 58.33 68.75 P3 58.33 75.00 79.58 P4 75.00 66.67 75.00 P5 81.11 82.22 66.67 P6 66.67 68.75 81.11 L1 58.33 58.33 87.22 L2 81.11 73.33 75.00 L3 41.67 41.67 58.33 L4 62.50 62.50 72.78 L5 68.89 66.67 63.33

Table 10

The performance matrix [xij]mnwith the best value xj and the worst value xj by VIKOR.

Indexes C Bank S Bank U Bank x

j xj F1a 93.33 87.22 79.58 93.33 79.58 F2b 66.67 75.00 68.75 66.67 75.00 F3a _{50.00 50.00 66.67 66.67 50.00} F4a _{41.67 58.33 75.00 75.00 41.67} F5a 62.50 68.75 73.33 73.33 62.50 F6a 56.25 62.50 87.23 87.23 56.25 C1a _{66.67 56.27 58.33 66.67 56.27} C2a _{72.80 41.67 37.50 72.80 37.50} C3a 87.23 50.00 75.00 87.23 50.00 C4a 50.00 58.33 41.67 58.33 41.67 C5a 73.33 70.00 66.67 73.33 66.67 C6a 75.00 62.50 87.23 87.23 62.50 P1a 68.75 62.50 50.00 68.75 50.00 P2a 62.50 58.33 68.75 68.75 58.33 P3b 58.33 75.00 79.58 58.33 79.58 P4a 75.00 66.67 75.00 75.00 66.67 P5a 81.11 82.22 66.67 82.22 66.67 P6a _{66.67 68.75 81.11 81.11 66.67} L1b 58.33 58.33 87.22 58.33 87.22 L2a 81.11 73.33 75.00 81.11 73.33 L3a 41.67 41.67 58.33 58.33 41.67 L4a _{62.50 62.50 72.78 72.78 62.50} L5a _{68.89 66.67 63.33 68.89 63.33} x

j indicates the best values for setting all the criteria functions (aspired/desired levels) and x

j indicates the worst values (tolerable/worst levels). a

Indicates the evaluation index is associated with benefit criteria and maximum is the ideal solution.

b

Indicates the evaluation index associated with cost criteria and minimum is the ideal solution.

Table 6

The normalized performance matrix [rij]mnof three banks by various experts.

Indexes C Bank S Bank U Bank

F1 1.00 0.56 0.00 F2 1.00 0.00 0.75 F3 0.00 0.00 1.00 F4 0.00 0.50 1.00 F5 0.00 0.58 1.00 F6 0.00 0.20 1.00 C1 1.00 0.00 0.20 C2 1.00 0.12 0.00 C3 1.00 0.00 0.67 C4 0.50 1.00 0.00 C5 1.00 0.50 0.00 C6 0.51 0.00 1.00 P1 1.00 0.67 0.00 P2 0.40 0.00 1.00 P3 1.00 0.22 0.00 P4 1.00 0.00 1.00 P5 0.93 1.00 0.00 P6 0.00 0.14 1.00 L1 1.00 1.00 0.00 L2 1.00 0.00 0.21 L3 0.00 0.00 1.00 L4 0.00 0.00 1.00 L5 1.00 0.60 0.00 Table 7

The weighted normalized performance matrix [vij]mnwith the ideal solutions A*and the negative ideal solutions A_{by TOPSIS.}

Indexes C Bank S Bank U Bank A*

A F1a 0.0604 0.0335 0.0000 0.0604 0.0000 F2b _{0.0309 0.0000 0.0232 0.0000 0.0309} F3a _{0.0000 0.0000 0.0812 0.0812 0.0000} F4a 0.0000 0.0392 0.0784 0.0784 0.0000 F5a 0.0000 0.0237 0.0410 0.0410 0.0000 F6a 0.0000 0.0107 0.0532 0.0532 0.0000 C1a _{0.1237 0.0000 0.0246 0.1237 0.0000} C2a 0.0715 0.0084 0.0000 0.0715 0.0000 C3a 0.0527 0.0000 0.0354 0.0527 0.0000 C4a 0.0370 0.0741 0.0000 0.0741 0.0000 C5a 0.0371 0.0185 0.0000 0.0371 0.0000 C6a 0.0198 0.0000 0.0392 0.0392 0.0000 P1a 0.0438 0.0292 0.0000 0.0438 0.0000 P2a 0.0093 0.0000 0.0232 0.0232 0.0000 P3b 0.0158 0.0034 0.0000 0.0000 0.0158 P4a 0.0242 0.0000 0.0242 0.0242 0.0000 P5a _{0.0113 0.0121 0.0000 0.0121 0.0000} P6a _{0.0000 0.0019 0.0128 0.0128 0.0000} L1b 0.0109 0.0109 0.0000 0.0000 0.0109 L2a 0.0383 0.0000 0.0082 0.0383 0.0000 L3a 0.0000 0.0000 0.0245 0.0245 0.0000 L4a _{0.0000 0.0000 0.0246 0.0246 0.0000} L5a 0.0263 0.0158 0.0000 0.0263 0.0000 A*

indicates ideal solutions and A

indicates negative ideal solutions. a

Indicates the evaluation index is associated with benefit criteria and maximum is the ideal solution.

b

Indicates the evaluation index associated with cost criteria and minimum is the ideal solution.

Table 8

The separations of the ideal solution d

iand the negative-ideal solution d  i by TOPSIS. Banks di di C Bank 0.1480 0.1867 S Bank 0.1982 0.1077 U Bank 0.1731 0.1537 Table 9

The relative closeness RC

i to the ideal solution and preference order ranking by TOPSIS.

Banks C Bank S Bank U Bank

RC

i 0.5579 0.3521 0.4704

Ranking 1 3 2

Note: The ranking is based on the relative closeness indicating the gap to improve for achieving the aspired/desired level. The largest value means the smallest gap to achieve the ideal level.

Table 11

The values Siand Riby VIKOR.

Banks Si Ri

C Bank 0.0000 (1) 0.0000 (1)

S Bank 1.0000 (3) 1.0000 (3)

U Bank 0.3599 (2) 0.4219 (2)

Note: () indicates ranking order.

Table 12

The value Qiwithv= 0, 0.5, 1 and preference order ranking by VIKOR for sensitive analysis.

Banks Qi[v= 0] Qi[v= 0.5] Qi[v= 1]

C Bank 0.0000 (1) 0.0000 (1) 0.0000 (1)

S Bank 1.0000 (3) 1.0000 (3) 1.0000 (3)

U Bank 0.3599 (2) 0.3909 (2) 0.4219 (2)

4.4. Discussion

This research conducted a performance analysis on three banks using a FMCDM approach based on the BSC perspectives. The FAHP and the three MCDM analytical methods (i.e. SAW, TOPSIS, and VI-KOR) were employed in the performance analysis for computing the fuzzy weights of the criteria, ranking the banking performance and improving the gaps of the three banks, respectively. Based on the results of the analysis, some essential findings were discussed as follows.

The FAHP adopted by the research, which combines the AHP with fuzzy set theory, can not only capture the thinking logic of hu-man beings but also focuses on the relative importance of the

eval-uation criteria of the banking performance. As shown inTable 2,

the result of the FAHP analysis reveals that the ‘‘Customer” dimen-sion is the primary focus of the BSC and ‘‘customer satisfaction” is the most important evaluation index. This is because banking is a service industry, and banking performance is strongly connected to customer satisfaction. Therefore, in addition to paying attention to the financial indexes, such as ROA and EPS, which are ranked as the second and third most import indexes for sustaining a high banking performance, banks also must ensure that their customers remain loyal to them and develop new markets to attract new customers.

In addition, as mentioned in Section 3.4, the TOPSIS method

is used to provide information on how to improve the gaps in each criteria so as to achieve the bank’s objective

(desired/as-pired level) and cannot be used for ranking purpose (Opricovic

& Tzeng, 2004). Therefore this research adopted the VIKOR method for ranking and improving the alternatives of this prob-lem. Hence, based on the fuzzy weights of the evaluation criteria calculated by FAHP, the performance ranking order of the three banks using SAW is C Bank  U Bank  S Bank. The ranking or-der is the same as the results or-derived from both TOPSIS and VI-KOR. However, the result found that it was evident that the final values calculated by VIKOR distinguished the banking perfor-mance among the three banks. This finding is consistent with previous studies (Chu, Shyu, Tzeng, & Khosla, 2007; Opricovic & Tzeng, 2007).

When comparing the performance of S Bank with that of the

other two banks, as shown inTable 5, it is evident that S Bank

has the poorest performance value in the ‘‘Customer” dimension while this is the most important BSC factor according to the ex-perts. As far as the evaluation indexes within the ‘‘Customer” dimension are concerned, S Bank has the lowest performance va-lue in the ‘‘market share rate” index. This implies that increasing its market share must be considered a crucial factor in that bank’s growth strategy. Therefore, in addition to retaining its existing customers, S Bank should also develop new service items and/or provide more and improved promotions to attract new customers in order to keep up with the other two banks. Based on the performance analysis, it is evident that the main reason for S Bank being ranked lowest is due to the fact that its performance values from a customer perspective are poor. Therefore, for S Bank to improve its performance, it must first put more emphasis on customer satisfaction, and then on finan-cial return.

5. Conclusions and remarks

In response to the rapid growth of service industries and the in-creased global competition, particularly for the banking

institu-tions, the need for alternative controls and performance

measures has attracted much attention. However, researchers are finding it difficult to measure banking performance because of the intangible nature of the products and services of the banking industry. According to the relevant literature, most studies only

used financial factors to evaluate banking performance (Kosmidou

et al., 2006). The present research proposed a FMCDM evaluation model for banking performance by determining a comprehensive set of evaluation criteria based on the concept of the BSC. Our pro-posed model embraces both financial and non-financial aspects, and optimizes the relationships of a bank. It matches the needs and requirements of the clients with the delivery processes of the bank in order to achieve the best possible customer satisfaction through effective performance.

Based on the extensive content of the BSC evaluation criteria for banking performance as selected from the relevant literature and the objective opinions synthesized from the experts, the FAHP and the three MCDM analytical methods (i.e. SAW, TOPSIS, and VIKOR) were adopted in the performance analysis for com-puting the fuzzy weights of the criteria, and for ranking the banking performance of three banks as an illustrative example. The relative fuzzy weights calculated by FAHP prioritize the importance of the BSC evaluation criteria for banking perfor-mance. With respect to the relative weights of the criteria, it not only reveals the ranking order of the banking performance but it also pinpoints the gaps to better achieve the bank’s goal by using the MCDM analytical methods. The analysis result indi-cates that management should make good use of the limited re-sources available to improve those aspects of their business that needs improvement the most. Our proposed framework with FMCDM shows to be a feasible and effective assessment model for banking performance evaluation, and it can be applied to other institutions as well.

In conclusion, the findings of this study can be summarized as follows: 1. Integrating all the relevant experts’ opinions, 23 out of the 55 evaluation indexes are selected as being suitable for bank-ing performance in terms of BSC perspectives; 2. By applybank-ing the FAHP, the order of relative importance of the four BSC perspec-tives for banking performance is ‘‘C: Customer”, ‘‘F: Finance”, ‘‘L: Learning and growth”, and ‘‘P: Internal process”. The top five pri-orities of the evaluation indexes are ‘‘C1: Customer satisfaction”, ‘‘F3: Return on assets”, ‘‘F4: Earnings per share”, ‘‘C4: Customer retention rate, ” and ‘‘C2: Profit per customer”, respectively; and 3. Using the fuzzy weights of the criteria calculated by FAHP, the ranking of the banking performance of the three banks by employing the MCDM analytical methods is U Bank, C Bank, and S Bank, respectively. Based on our findings the following sugges-tions are made. First, since there is no one performance evalua-tion index to fit all, performance evaluaevalua-tion indexes should be tailored to meet the organization’s overall goals as well as the objectives of each individual unit. Second, the performance evalu-ation indexes of the BSC perspectives may not be mutually inde-pendent. Other analytical methods (e.g. fuzzy integral, Analytic Network Process, etc.) can be employed to solve the interactive and feedback relations among indexes. Third, future research may utilize several other techniques to investigate the casual relationships among performance evaluation indexes of the BSC to objectively build strategy maps. Finally, exploring more cases and conducting more empirical studies are recommended to fur-ther validate the usefulness of the proposed performance evalua-tion model.

Table 13

Summary of final values and preference order ranking by three methods.

Banks SAW TOPSIS VIKOR

C Bank 113.97 (1) 0.5579 (1) 0.0000 (1)

S Bank 107.99 (3) 0.3521 (3) 1.0000 (3)

U Bank 113.65 (2) 0.4704 (2) 0.3909 (2)

Appendix. Descriptions of the selection evaluation indexes for banking performance

No. Selection evaluation

indexes

Description

1 (F1) Operating

revenues

Sales revenue

2 (F2) Debt ratio Debt divided by assets

3 (F3) Return on assets

(ROA)

After-tax profit/ loss divided by average total assets

4 (F4) Earnings per

share (EPS)

After-tax net earning minus

preferred share dividends divided by weighted average number of shares outstanding

5 (F5) Profit margin After-tax profit/ loss divided by total

operating revenues

6 (F6) Return on

investment (ROI)

After-tax profit/ loss divided by total cost

7 (C1) Customer

satisfaction

Customer satisfaction of products and service

8 (C2) Profit per

on-line customer

After-tax earnings divided by total number of on-line customers

9 (C3) Market share

rate

Sales volumes of products and services divided by total market demands

10 (C4) Customer

retention rate

Capability of keeping existing customers

11 (C5) Customer

increasing rate

Growth rate of new customers

12 (C6) Profit per

customer

After-tax earnings divided by total number of customers

13 (P1) No of new

service items

Total numbers of new service items

14 (P2) Transaction

efficiency

Average time spent on solving problems occurring during transactions

15 (P3) Customer

complaint

Customer criticisms due to dissatisfaction about products and services

16 (P4) Rationalized

forms and processes

Degree of procedures systemized by documentations, computer software, etc.

17 (P5) Sales

performance

Successful promotion of both efficiency and effectiveness of sales

18 (P6) Management

performance

Improvement of effectiveness, efficiency, and quality of each objective and routine tasks

19 (L1) Responses of

customer service

Numbers of suggestions provided by customers about products and services

20 (L2) Professional

training

Numbers of professional

certifications or training programs per employee 21 (L3) Employee stability Turnover of employees 22 (L4) Employee satisfaction

Employee satisfaction about both hardware and software provided by the company

23 (L5) Organization

competence

Improvement of project management, organizational capability, and management by objectives (MBO), etc.

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