Table 7Final relationship matrix between WHATs and HOWs with both crisp and STFNs.

Final matrix

WHATs

HOWs

L Inventory management U L Total quality management U L Human resource management

U

W1 4.1 5.1 6.1 5.5

6.5 7.5 6.2 7.2 8.2

W2 3.7 4.7 5.7 4.4

5.4 6.4 5.6 6.6 7.6W3 4.1 5.1 6.1 4.

55.5 6.5 4.8 5.8 6.8

W4 5.0 6.0 7.0 3.7

4.7 5.7 4.0 5.0 6.0W5 4.7 5.7 6.7 4.

75.7 6.7 4.2 5.2 6.2

L Knowledge management U L Technology management U L Information management

U

W1 4.8 5.8 6.8 4.5

5.5 6.5 5.1 6.1 7.1W2 4.6 5.6 6.6 4.

15.1 6.1 5.1 6.1 7.1

W3 4.1 5.1 6.1 5.1

6.1 7.1 4.4 5.4 6.4W4 4.2 5.2 6.2 5.

36.3 7.3 4.8 5.8 6.8

W5 4.9 5.9 6.9 5.0

6.0 7.0 4.4 5.4 6.4

L Energy management U L Project management U L Financial management UW1 4.5 5.5 6.5 3.

94.9 5.9 5.3 6.3 7.3

W2 4.6 5.6 6.6 4.0

5.0 6.0 4.5 5.5 6.5W3 3.4 4.4 5.4 4.

05.0 6.0 4.8 5.8 6.8

W4 4.2 5.2 6.2 4.0

5.0 6.0 5.0 6.1 7.1W5 4.0 5.0 6.0 3.

84.8 5.8 5.0 6.0 7

L Change management U L Customer relationship management

U L Supply chain management

UW1 4.6 5.6 6.6 4.

55.5 6.5 4.4 5.4 6.4

W2 4.7 5.7 6.7 4.2

5.2 6.2 3.6 4.6 5.6W3 4.8 5.8 6.8 5.

16.1 7.1 4.3 5.3 6.3

W4 4.6 5.6 6.6 4.9

5.9 6.9 5.0 6.0 7.0W5 5.0 6.0 7.0 4.

15.1 6.1 4.0 5.8 6.8

L Business process management

U L Strategic management U L Production management

U

W1 5.1 6.1 7.1 4.7

5.7 6.7 4.0 5.0 6.0

W2 4.3 5.3 6.3 3. 4.9 5.9 4.2 5.2 6.2W3 5.1 6.1 7.1 5.

56.5 7.5 5.2 6.2 7.2

W4 5.5 6.5 7.5 4.3

5.3 6.3 6.0 7.0 8.0

W5 5.0 6.0 7.0 4.8

5.8 6.8 6.0 7.0 8.0

HOW's relationships with the WHATs are always examined once

the HOW is considered. The relationships between the HOWs and

the WHATs are determined by technical analysis and empirical

judgment, and usually may not be precise. So it is quite

appropriate to use STFNs to represent this kind of relationships.

For each HOW with respect to each WHAT, the experts determine

the relationship first in linguistic term using scale (4) and then

convert this relationship into corresponding crisp number and

STFN, for example, the expert consider the relationship between

H1 and W1 as ‘‘very strong’’ that corresponds to a crisp number of

9 and an STFN [8,10].

The full matrix of these relationships, both in crisp numbers

and STFNs, are shown in Table 7 where be obtained by averaging

the expert’ assessments about the relationship between WHATs

and HOWs.

Step 7: According to the WHATs' final importance ratings and

the relationship values between the HOWs and the WHATs, the

HOWs' initial technical ratings can be computed usually through

the simple additive weighting (SAW) formula (5) . When crisp

numbers are used, the initial technical ratings are given as

t = ... , t15)

= (1.38,1.35,1.40,1.38,1.48,1.42,1.27,1.23,1.49,1.44,1.39,1.40,1.51,1.41,1.61)

Here, for example, crisp initial technical rating ofH1, t1, is

computed as the weighted average over H1's crisp relationship

values with the

five WHATs, r11, r21.......r51, which correspond to the crisp part of

the

relationship matrix that is bolded in Table 7 , and the weights are

the crisp final importance ratings of the five WHATs, f1, f2.....f5,

i.e.,5

t1 = '^2 fm X rm1m=1

= 0.0286 x 5.1 + 0.02807 x 4.7 + 0.0369 x 5.1 + 0.068 x 6.0 +

0.087 x 5.7 = 1.38

From these crisp initial technical ratings, the technical measures

(HOWs) can be ranked in the following order:

H15 > H13 > H9 > H5 > H10 > H6 > H14 > H12 > H11 > H3

> H1 > H4 > H2 > H7 > H8 (9)

If fuzzy numbers of the relationship matrix are used, the fuzzy

initial technical ratings are also given as STFNs:

tf = ([0.73,2.52], [0.72,2.46], [0.75,2.52], [0.73,2.51], [0.8,2.67],

[0.76,2.57], [0.66,2.33], [0.64,2.28], [0.8,2.68], [0.77,2.60],

[0.74,2.52], [0.74,2.55], [0.82,2.72], [0.76,2.56],

9642 S. Yousefie et al./Expert Systems with Applications 38 (2011) 9633-9647

Table 8The crisp and fuzzy initial technical rating of HOWs.

HOWs

L Inventory management U L Total quality management U L Human resource management

Uf 1.38 1.35 1.4

if 0.73 2.52 0.72 2.46 0.75

2.52

L Knowledge management U L Technology management U L Information management Uf 1.38 1.48 1.42

tf 0.73 2.5 0.79 2.6 0.76

2.56

L Energy management U L Project management L Financial management Uf 1.26 1.23 1.49

tf 0.66 2.33 0.64 2.2 0.80

2.68

L Change management U L Customer relationship management

U L Supply chain management Uf 1.43 1.39 1.40tf 0.77 2.60 0.74 2.52 0.7

42.55

L Business process management U L Strategic management U L Production management U

t 1.51 1.41 1.61

tf 0.82 2.72 0.76 2.56 0.88 2.89

Table 9

Final technical competitive analysis matrix Y = [yni]15x7.

HOWs C1 C2 C3 C4 C5 C6 C7

Strategic management 5.56 5.00 4.43 4.96 5.40 4.50 6.43

Business process management 6.30 5.06 3.86 5.20 5.33 4.66 6.20

Supply chain management 6.53 5.30 5.03 5.33 5.56 4.76 6.66

Customer relationship management 6.13 5.46 5.10 5.20 5.60 4.33 6.70Change management 5.20 4.73 4.76 4.60 4.66 4.33 6.36Financial management 6.03 4.43 4.86 5.23 5.33 4.73 6.66

Project management 5.63 5.03 4.80 4.83 5.10 5.00 6.56Energy management 5.16 4.4 4.70 4.86 5.3 4.63 6.13Information management 5.13 4.86 4.66 5.13 4.96 4.6 6.9Technology management 4.80 4.70 4.63 5.00 4.93 4.4 6.46Knowledge management 5.03 5.10 4.50 5.03 4.93 4.86 6.53Human resource management 5.76 4.83 4.53 5.33 5.50 4.76 6.60Total quality management 5.70 5.16 4.23 4.86 5.23 4.56 6.53Inventory management 5.36 4.83 4.96 4.90 5.16 4.93 6.23Production management 6.46 5.23 5.03 5.26 5.23 5.13 7.03

mance on the HOWs, company C1 must try all the means to obtain

this valuable information in order to know its technical strengths

and weaknesses and hence to improve or enhance its

competitiveness. Through a lot of efforts company Ci obtains all

the technical parameters of its own and its competitors in terms

of the i5 HOWs. This information forms a technical comparison

matrix Y = [ynl]15x7 as shown in Table 9 .

Applying entropy method to Y in the same manner as in excel-

lence competitive analysis (Step 3), technical competitive priority

ratings can be obtained for company C1 on the 15 HOWs:

z = (Zi,Z2, . . . ,Zl4,Z15)

= (0.071431, 0.071285,0.071443,..., 0.07157,0.071409)

From these ratings which we know that H9; H15 and H5 are of the

highest competitive priorities. According to the technical perfor -

mance of its own and the other six competitors company in terms

of the 15 HOWs, company C1 could set technical performance goal

on each of the HOWs for itself.

To better fulfill the customer needs. These goals should be

determined both competitively and realistically. Company C 1’s rel-

evant experts agree with the following performance goals on the

HOWs for further improvement:

b = (i>1, b2,..., ^14, ^15) = (7, 8,8,8,7,8,7,7,7,6,7,7,7,7,8)

From these goal (bn) and current (yn1) technical performance

levels, improvement ratios for company C1 to be competitive in

terms of the HOWs can be easily computed using (6) as:

Here, for example, the initial technical rating of H1 in STFN form,

tf, is computed as the weighted average over H 1’s STFN form

relationship values with the five WHATs, r{1,r{1,..., r51, which

correspond to the first column of the STFN form relationship

matrix Rf, and the weights are the final importance ratings of the

five WHATs in STFN form, f{,f.. .f, i.e.,

tf1=X fmx 4m=1

= [0.016,0.0308]x[4.1,6.1]n-------+ [0.05,0.134]x[4.7,6.7]= [0.73,2.52]

According to the principle in the Appendix, these fuzzy ratings

have the following ranking order for the HOWs’ initial importance:

H15 > H13 > H9 > H5 > H10 > H6 > H14 > H12 > Hn > H3

> H1 > H4 > H2 > H7 > H8 (10)It is noticed from (9) and (10) that the crisp and fuzzy ratings

exhibit the same ranking order. Both sets of ratings indicate that

H15 is of the highest initial importance, followed by H13, H9 and H5.

The crisp and fuzzy initial technical ratings of the 15 HOWs are

shown in Table 8.

Step 8: Now turn to technical competitive analysis which is to

find and establish competitive advantages or to further enhance

the existing advantages for company C1, through comparing all

the company’ similar products in terms of their technical perfor -

mance on the 15 identified HOWs. Although it is always not easy

to acquire the technical performance levels of competitors’

perfor

S. Yousefie et at./Expert Systems with Applications 38 (2011) 9633-9647 9643

v = (v 1, v 2, ..., v 14, v 15)

= (1.25749,1.26984,..., 1.30435,1.23711)

Step 9: This is the last step of our proposed HOE model. Integrat -

ing the initial technical ratings, technical competitive priority rat-

ings and improvement ratios of the HOWs, final technical ratings

can be computed by (7) . If initial technical ratings are crisp num-

bers, the final technical ratings are also crisp numbers and given

as:

s = (s1, s2,..., s14, s«) = (0.12411,0.12247,..., 0.13315,0.13205)

Here, for example, the final technical rating of H 1 in crisp form, s1,

is computed by (7) as:

s1 = v 1 x t1 x Z1 = 1.25749 x 1.38 x 0.071431 = 0.12411

From s we can rank the final technical importance of the nine

HOWs in the following order:

H9 > H15 > H5 > H11 > H6 > H13 > H14 > H10 > H4 > H1

> H3 > H2 > H12 > H8 > H7 (11)

This final technical importance order differs from the initial

technical importance order (9) in two aspects: (i) H15 is of higher

initial technical importance but lower final technical importance

than H9; and (ii) H8 is of lower initial technical importance but

high final technical importance than H7. Since technical

competitive priority ratings (zn's) do not vary too much, these two

differences are mainly

Table 10Crisp and fuzzy final technical ratings of the 15 HOWs.

HOWs Final technical ratings Scaled final technical ratings

Crisp (sn) Fuzzy (s{,) Crisp (sn)

Fuzzy (sfn)

H1 0.124113 [0.066, 0.227]

0.854326

[0.252, 0.868]

h2 0.122475 [0.065,0.223]

0.843047

[0.249,0.853]H3 0.122697 [0.066,0.22

0]0.844581

[0.252, 0.846]H4 0.128393 [0.068,0.23

3]0.883786

[0.261,0.897]H5 0.142687 [0.077,0.25

7]0.982174

[0.294,0.987]H6 0.134406 [0.072,0.24

3]0.925178

[0.276,0.930]H7 0.112604 [0.059,0.20

7]0.77510 [0.225, 0.794]

Hg 0.119689 [0.062, 0.221]

0.82387 [0.238, 0.847]H9 0.145276 [0.079,

0.261]1.00000 [0.301, 1.000]

H10 0.128473 [0.069,0.232]

0.884335

[0.264,0.889]H11 0.138175 [0.074,

0.251]0.951116

[0.282,0.959]H12 0.121494 [0.065,

0.221]0.836298

[0.247, 0.847]H13 0.133152 [0.072,0.23

9]0.916544

[0.276, 0.915]H14 0.132053 [0.071,0.23

9]0.908977

[0.271, 0.914]H15 0.142942 [0.078,0.25

6]0.983933

[0.299,0.980]

caused by the setting of performance goals (b n’s) or improvement

ratios (vn’s): (i) H15’s improvement ratio (1.23711) is lower than

H9’s (1.3636), and (ii) H8’s improvement ratio (1.3548) is higher

than H7’s (1.2426).

If initial technical ratings are STFNs, then the final technical

ratings are also given as STFNs:

Sf = {Sfl, ... s15)

= ([0.066,0.227], [0.065,0.223],..., [0.071, 0.239], [0.078,0.256]).

Here, for example, the final technical rating of H1 in STFN form,

sf1, is computed by (7) and the arithmetic of STFNs as:

= v1 x tf1 x z1 = 1.25749 x [0.73, 2.52] x 0.071431

= [0.066, 0.227]

These fuzzy ratings produce the following ranking order for the

HOWs' final importance:

H9 > H15 > H > H11 > H6 > H13 > H14 > H10 > H4

> H1 > H3 > H2 > H12 > H8 > H7. (12)

It is noticed from (11) and (12) that the crisp and fuzzy ratings

show an almost identical ranking order for the HOWs' final

technical importance. Both sets of ratings indicate that H9 is the

most important HOW, followed by H15 and then by H11 and H6, and

that H7 is the least important HOW, preceded by H38 and H12.

These crisp and fuzzy final technical ratings of the HOWs are

shown in Table 10 . In order to be comparable, they are both

scaled to have maximum rating or upper limit of unity, which are

also shown in Table 10 . From these scaled ratings we can see

again that, although the crisp and fuzzy ratings exhibit an

identical trend, crisp ratings always tend to be close to the upper

limits of the corresponding fuzzy ratings. This shows that fuzzy

ratings are more representative of the possible variations of the

HOWs’ technical importance, which would make the technical

improvement more flexible and the design process more feasible.

The above nine steps complete the HOE process for improving

the company's improvement trends on excellence. The corre-

sponding tables of results, after appropriate arrangement, can

form an HOE like Fig. 2 which links organization needs to tech-

nical considerations and exhibits all the relevant elements and

their relationships. As a result of this HOE model, it is concluded

that H7 could be deleted from further consideration (in QFD’s

second phase, parts deployment) to save technical efforts without

decreasing organization satisfaction. If resource or budget

considerations require to further cut down the number of HOWs,

Table 11Normalization and determine the percentages of the 15 HOWs.

HOWs Crisp (sn) Fuzzy (sfn)

Crisp weights Nor. Per. % Fuzzy weights Defuzzi. Nor. Per. %

H1 0.124113 0.063693 6.36926 [0.066,0.227] 0.135156 0.063825 6.38254

H2 0.122475 0.062852 6.28517 [0.065,0.223] 0.133169 0.062887 6.288714H3 0.122697 0.062966 6.29661 [0.066,0.220] 0.13305 0.062831 6.283073H4 0.128393 0.065889 6.58889 [0.068,0.233] 0.139687 0.065965 6.59649H5 0.142687 0.073224 7.32241 [0.077,0.257] 0.154875 0.073137 7.313718H6 0.134406 0.068975 6.89749 [0.072,0.243] 0.145948 0.068922 6.892164H7 0.112604 0.057786 5.77861 [0.059,0.207] 0.122847 0.058013 5.801264h8 0.119689 0.061422 6.1422 [0.062,0.221] 0.130701 0.061721 6.172143H9 0.145276 0.074553 7.45531 [0.079,0.261] 0.157568 0.074409 7.440888H10 0.128473 0.06593 6.59299 [0.069,0.232] 0.139532 0.065892 6.589179H11 0.138175 0.070909 7.09086 [0.074,0.251] 0.150141 0.070902 7.090178H12 0.121494 0.062349 6.23485 [0.065,0.221] 0.132208 0.062433 6.243329H13 0.133152 0.068331 6.83312 [0.072,0.239] 0.144356 0.06817 6.816961H14 0.132053 0.067767 6.7767 [0.071,0.239] 0.143414 0.067725 6.772494H15 0.142942 0.073355 7.33552 [0.078,0.256] 0.154942 0.073169 7.316866

9644 S. Yousefie et al./Expert Systems with Applications 38 (2011) 9633-9647

H8; H12 and H2 form a good deleting order that will not significantly

influence the fulfillment of the organization needs. And also,

according to the Table 10 , by use of these crisp and fuzzy

importances, the importance weighting of each management tool

can be computed. In this way, first of all, the crisp importance

values of management tools should be normalized, and then the

percentages of them should be calculated. But for the fuzzy

importance values, at first, the defuzzification of the fuzzy values

of management tools should be determined, for this step has

done by the Facchinetti et al. (1998) approach, and then the nor-

malization and determine the percentage of them, should be done

(see Table 11 ).

Hypothesize 2: The ranking results of effective management

tools on setting EFQM excellence model in an organization

(the research case) are the same in crisp and fuzzy

approaches.

In order to be comparable, the crisp and fuzzy final importance

ratings are tested by spearman correlation coefficient. According

to the Table 11 , with percentage results of crisp and fuzzy

ranking, the spearman coefficient correlation for these tow type of

data is 0.993 and there is a very strong positive correlation

between fuzzy and crisp importance ranking. So the first

hypothesis of this research that maintains: ‘‘The ranking results of

effective management tools on setting EFQM excellence model in

an organization (the research case) are the same in crisp and

fuzzy approaches.’’, were supported.

5. Conclusions

Using the useful management tools that are relevant to the

organization's needs for excellence has become so important. By

choosing and applying the best management tools among too

many management tools, companies can improve their perfor-

mances and then increase customer satisfaction and gain market

shares. But for the organizations, that adopted excellence models

such as EFQM, to improve their performances, selection and

choosing these management tools has been a big challenge in

today's dynamic environment. This paper presents a systematic

and operational approach to HOE to help resolve this problem.

This study has addressed the applicability of QFD in the

organizational excellence context. More specifically, an original

methodology has been proposed and adopted to rank viable EFQM

excellence criteria and the management tools a firm can

undertake to improve excellence performances.

The methodology developed could be rightly considered as a

useful tool for selecting the most efficient and effective manage-

ment tools leverages to reach organizational excellence. We pro-

pose a 9-step HOE model, which is basically a QFD model, to unify

the HOE process and a few 9-point scales to unify the mea-

surements in HOE to avoid arbitrariness and incomparability.

We especially address the various ‘‘voices’’ in the HOE

process and suggest the use of symmetrical triangular fuzzy

numbers (STFNs) to reflect the vagueness in expert’s linguistic

assessments. Furthermore, we employ the quantitative entropy

method to conduct competitive analysis and derive competitive

priority ratings. All information required, computations involved

and feasible methods are clearly indicated to give an applicable

framework for practitioners to perform HOE analysis without

confusions and difficulties. To fully illustrate our proposed HOE

model, we present an automotive company example that involves

five organizational needs for excellence (EFQM enabler criteria),

15 technical attributes for excellence (management tools) and

seven competitor companies.

In a similar manner, the weighted importance of management

tools allows the firm to identify the key factors of intervention in

order to improve the perceived excellence. As an example, pro

Production management (7.32)

Fig. 4. The basis for programming and organizational resources allocation for excellence.

S. Yousefie et al./Expert Systems with Applications 38 (2011) 9633-9647 9645

cesses emerges in step 4 as the most important factor from

experts’ point of view, and it should be considered as the key

excellence criterion to improve the performance of the

organization. In order to assess and rank viable management

tools, in the approach proposed we have introduced entropy

method, which considers the competition of implementation for

each ‘‘what’’ and ‘‘how’’. The entropy can be directly adopted as

a synthesis parameter to select the most suitable EFQM enabler

criteria and management tools that have the most competitive

importance to implement. According to step 3 and step 8, it is

considered that WHATs and HOWs, both of them have the same

competitive importance. Since personal judgments are required

when building the HOE, fuzzy logic has been adopted as a useful

tool. Through fuzzy logic linguistic judgments an expert gives to

weights, relationships and correlations have been appropriately

translated into triangular a fuzzy number. Moreover, fuzzy logic

has allowed to cope well with uncertainties and incomplete

understanding of the relationships between ‘‘WHATs’’ and

between ‘‘HOWs’’ and ‘‘WHATs’’. In addition, fuzzy logic becomes

fundamental to dealing with several parameters that seem

difficult to express in a quantitative measure. As an example,

detailed information about relationships between management

tools and EFQM excellence criteria are usually not available, while

linguistic judgments on them can be easily obtained.

By use of the fuzzy importance percents ranking of EFQM crite-

ria and management tools, from Tables 6 and 11, the basis for

programming and allocating of organization resources for the

improving of excellence performances, can provided. It is shown

in Fig. 4. The methodology proposed does not deal with the practi -

cal implementation of management tools. Future work may be

thus directed to extend a similar QFD approach from a strategic

level to tactical and operational ones. Specially, future work can

extend sub-set of each management tools in to the other phases

of QFD approach.

Appendix A

A.1. Fuzzy methods

Fuzzy set theory was developed for solving problems in which

descriptions of objects are subjective, vague and imprecise, i.e.,

no boundaries for the objects can be well defined. Let X = {x} be

a traditional set of objects, called the universe. A fuzzy set E in X

is characterized by a membership function 1q(x) that associates

each object in X with a membership valueE in the interval [0, 1],

indicating the degree of the object belonging to E. A fuzzy number

is a special fuzzy set when the universe X is the real line R 1 : —i <

x < +i. A symmetrical triangular fuzzy number (STFN), denoted as

E = [0,1], is a special fuzzy number with the following

symmetrical triangular type of membership function:

1~E(x) = 1— I x — (c + a)/2 I /[(c — a)/2L a < x < c (A.1)

STFN is widely used in practice to represent a fuzzy set or

concept E = ‘‘approximately b’’ where b = (a + c)/2. For

example, if an EFQM enabler criterion leadership is rated as

having ‘‘very high’’ importance by a decision maker, then

traditionally we may assign leadership a number 9 using crisp

scale. To capture the vagueness of the decision maker’s

subjective assessment, we can according to the same scale

assign leadership an STFN [8,10] which means ‘‘approximately

9’’ and is represented by the following membership function:

This means that, for example, the membership value or

‘‘possibility’’ that leadership is assigned a number 9 is i[ 8,10](9) =

1, the ‘‘possibility’’ that leadership is assigned a number 8.5 or

9.5 is i[810](8.5) = 0.5 or i[810](9.5) = 0.5. So assigning leadership a

number 8.5 or 9.5 is acceptable or ‘‘possible’’ to the degree of

50%. The basic arithmetic rules for STFNs are as follows:

Addition : [a, b] + [c, d] = [a + c, b + d] (A.3)

Subtraction : [a, b] — [c, d] = [a — c, b — d] (A.4)

Scalar multiplication : k x[a, b] = [ka, kb] k > 0 (A.5)

Multiplication : [a, b]x[c, d]« [ac, bd], a > 0 c > 0 (A.6)

Division : [a, b]^[c, d]« [a/c, b/d], a > 0, c > 0 (A.7)

For any two STFNs, E1 = [a, b] and, E

2 = [c, d], if one interval is not

strictly contained by another then their ranking order can be

easily and intuitively determined. That is

• If d > b and c p a, or d p b and c> b, then E2 > E

1, where

means ‘‘is more importance or preferred than’’.

• If a = c, b = d, then E2 = E1

But if one interval is strictly contained by another, i.e., if d < b

and c > a, or d > b and c < a, then the ranking problem becomes

complex and many possibilities may occur. For more details

about fuzzy set theory, STFNs and fuzzy ranking methods, see

Zimmermann (1987) .

A.2. Fuzzy AHP

To apply the process depending on this hierarchy, according to

the method of Chang’s (1996) extent analysis, each criterion is

taken and extent analysis for each criterion, g,; is performed on,

respectively. Therefore, m extent analysis values for each

criterion can be obtained by using following notation (Kahraman,

Cebeci,

& Ruan, 2004 ):

M1- M2 M3 Mm gi’ gi’ gi’ . .. ’ gi

where g is the goal set (i = 1,2,3, 4, 5............n) and all the M’gi

(j = 1,2,3, 4, 5.....m) are triangular fuzzy numbers (TFNs). The

steps of Chang’s analysis can be given as in the following.

Step 1: The fuzzy synthetic extent value (Si) with respect to the

ith criterion is defined as Eq. (A.8)

Si = Z j

j—1

Mjgii=1 j=1

To obtain Eq. (A.9) ;

Mjgi

gij=1

(A.8)

(A.9)

Perform the ‘‘fuzzy addition operation’’ of m extent analysis

values for a particular matrix given in Eq. (A.10) below, at the

end step of calculation, new (l, m, u) set is obtained and used for

the next:

EMi =j=1

j=1

j=1

"j tujjj=1 ! (A.10)

1[8,10](x) = 1—|x — 9 I, 8 < x < 10. (A.2) where l is the lower limit value, m is the most promising value and

u is the upper limit value. And to obtain Eq. (A.11) ;

9646 S. Yousefie et al. /Expert Systems with Applications 38 (2011) 9633-9647

EEK>1 j=1

Perform the ‘‘fuzzy addition operation’’ of Mjgi(j = 1, 2, 3, 4, 5...m)

values give as Eq. (A.12) :

'jZZKi =i=1 j=1

n n n \

i=1 i=1 i=1 )

(A.12)

And then compute the inverse of the vector in Eqs. (A.12) and

(A.13) is then obtained such that

££M. i=1 j=1

ui; mi; 1/J2 li (A.13)

Step 2: The degree of possibility ofM2 = (t2, m2, u2) p M1 = (t1, m1, u1) is defined as Eq. (A.14) :

V(M2 > M1) = Sup[min(lM1(x), lM2(y))]Y>x

(A.14)

And x and y are the values on the axis of membership function

of each criterion. This expression can be equivalently written as

given in Eq. (A.15) below:

(1 if m2 p m1,

0 if I2 P u1, (A.15)Otherwise

To compare M1 and M2; we need both the values of V(M2 p M1)

and V(M1 p M2):

Step 3: The degree possibility for a convex fuzzy number to be

greater than k convex fuzzy numbers

Mi (i = 1, 2 , 3 , . . . , k) can be defined by V(M > M1,M2, . . . , Mk)

= V[(M > M1)&(M > M2)& •• • &(M > Mk)] = min V(M > Mi), i =

1,2, . . . , k

Assume that Eq. (A.16) is

dl(Ai) = min V(Si > Sk)(A.16)

For k = 1, 2, 3, 4, 5...n; k — i. Then the weight vector is given by

Eq. (A.17) :

Wl = (dl(A1), dl(A2) , . . . , dl(An))T

(A.17)

where Ai (i = 1, 2 , 3 , 4, 5, 6. .n) are n elements.

Step 4: Via normalization, the normalized weight vectors are

given in Eq. (A.18) :

W = (d(A1), d(A2) , . . . , d(An))T

(A.18)

where W is non-fuzzy numbers.

A.3. Entropy method for competitive priority ratingsw,

x= W

2

c,(Xn

c2 ■

Xu ■

■ CL

■ Xu\

X21 X22 • • X2L

WMXm ' • XMLJ

where xml is the performance of company C1’s on organizational

needs for excellence (ONE) Wm, perceived by the experts. Based

on this X information, the company C1 may set priorities on the M

organizational needs for excellence in order to achieve a relative

competitive advantage over other companies. If company C1 per-

forms much better than any other companies in terms of

organizational needs for excellence Wm, then further

improvement may not be urgently needed and thus a lower

priority could be assigned to Wm. At the other extreme, if C1

performs much worse than many other companies on Wm, then it

may be difficult for C1 to build a competitive advantage within a

short period of time. In both cases, Wm could be assigned a lower

priority rating. However, if most companies perform quite

similarly on Wm, not too much improvement effort from C1 may

result in a better performance of its excellence and give C1 a

unique competitive advantage. Thus a higher priority could be

assigned to Wm. In particular, if all companies’ performances on

Wm are the same, it implies a great excellence opportunity since

any improvement would create a significant competitive

advantage. So the highest priority could be assigned to Wm. This

basis of assigning priorities is interestingly related to the entropy

concept in information theory. Entropy is a measure for the

amount of information (or uncertainty, variations) represented by

a discrete probability distribution, p1, p2 pL:

E(W 1) = -Ul j>ml ln(Pml) (A.19)

where UL = 1/ln(L) is a normalization constant to guarantee 0 6

E(p1, p2.............pL) 6 1. Larger entropy or E(p1, p2........pL) value

implies smaller variations among the pl’s and hence less

information contained in the distribution. For the mth row of the

customer comparison matrix X corresponding to the

organizational needs for excellence need

Wm,Xm1,Xm2, ■ ■ ■ , X m l , let Xm = PXml be the total score

with respect to Wm. Then according to (A.19) , the normalized

ratings pml = xml/xm for l = 1 , 2 ....L can be viewed as the

‘‘probability distribution’’ of Wm on the L companies with

E(W 1) = -U^Pml ln(Pml) = E(W 1) l=1 1

= -U^Xml ln(Xml=Xm)

(A.20)

It is clear that the larger the E(Wm) value, the less information

contained in Wm or smaller variations among the pml’s (or xml’s). If

all

companies’ performance ratings on Wm, xm1, xm2..........xmL, are the

same, Wm has zero variations and E(Wm) achieves its maximum

of 1. So E(Wm) can be used to reflect the relative competitive

advantage in terms of the organizational needs for excellence

Wm. All these E(Wm) values, after normalization:

em = E(Wm) HT E(Wm), ,M (A.21)

em can be considered as the excellence competitive priority

ratings for company C1 on the M organizational needs for

excellence, with a larger em indicating higher competitive

priority for the corresponding Wm. For more on entropy and its

applications (Chan, Kao, Ng, & Wu, 1999 ).

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