O R I G I N A L P A P E R
Fuzzy decision maps: a generalization of the DEMATEL methods
Gwo-Hshiung Tzeng•Wen-Hsien Chen•Rachung Yu•Meng-Lin Shih
Published online: 29 October 2009 Springer-Verlag 2009
Abstract The Decision making trial and evaluation laboratory (DEMATEL) method is used to build and ana-lyze a structural model with causal relationships between different criteria. In this paper, it shows that DEMATEL is the specific case of fuzzy decision maps (FDM) when the threshold function is linear. Both FDM and DEMATEL have the same direct and indirect influence matrix. FDM incorporates the eigenvalue method, the fuzzy cognitive maps, and the weighting equation. In addition two numerical examples are illustrated to demonstrate the proposed results. On the basis of the mathematical proof and numerical results, we can conclude that FDM is a generalization of DEMATEL method.
Keywords Structural model Fuzzy decision maps Fuzzy cognitive maps Decision making trial and evaluation laboratory Multiple criteria decision making
1 Introduction
Making decisions is the part of our daily lives. The major consideration is that almost all decision problems involved multiple, usually conflicting, and interactive criteria. Many methods in multiple criteria decision making (MCDM) had developed to solve the problems (Chen and Hwang1992). These methods are based on multiple attribute utility theory (MAUT), have been proposed (e.g., the weighted sum and the weighted product methods) to deal with the MCDM problems. The concept of MAUT is to aggregate all criteria to a specific unique-dimension which is called utility function to evaluate alternatives. Although many papers have been proposed to discuss the aggregation operator of MAUT (Fishburn1970), the main problem of MAUT is the assumption of preferential independence (Grabisch 1995; Hillier 2001).
Utility independence or utility separability is usually the basic assumption of the multiple attribute decision making (MADM) methods for employing the additive function to represent the preferences of decision-makers. However, in the realistic problems, the assumption of utility indepen-dence or utility separability seems to be irrational. There-fore, it is interesting to clarify the structure among criteria, and then we can determine the appropriate MADM meth-ods based on the results of structural models.
Both Decision Making Trial and Evaluation Laboratory (DEMATEL) method and fuzzy decision map (FDM) could clarify the structure among criteria (Chih et al.2006; Huang and Tzeng 2007; Liou James et al. 2007; Yu and G.-H. Tzeng
Department of Business Administration, Kainan University, Taoyuan, Taiwan G.-H. Tzeng
Institute of Management of Technology,
National Chiao Tung University, Hsinchu, Taiwan W.-H. Chen
Department of Management Information System, Takming University of Science and Technology, Taipei, Taiwan
R. Yu (&)
Department of Information Technology, Ching Kuo Institute of Management and Health, Keelung, Taiwan
e-mail: ray@ems.cku.edu.tw M.-L. Shih
Department of Information Management, National Chung Cheng University, Ming-Hsiung, Taiwan
Tzeng 2006) for solving MCDM problems in preferential weights by using the AHP (independence) or ANP (dependence and feedback) or fuzzy integral (inter-depen-dent/relation). The assumption of the DEMATEL method is that it’s the direct and indirect influence matrix is generating by linear transformation. Comparing to the DEMATEL, the FDM will have better fit in the real world situation due to the flexible threshold function. In this paper, it uses math-ematical proof to show that FDM is a generalization method of the DEMATAL. Two numerical examples are illustrated to demonstrate the proposed results.
The rest part of this paper is organized as follows. In Sect.2, we describe the contents of the DEMATEL method. FDMs and mathematical proof are proposed in Sect.3. Two numerical examples, which are used here to demonstrate the proposed results, are in Sect.4. Discussions are presented in Sect.5and conclusions are in last section.
2 The DEMATEL method
The DEMATEL method, developed by the Science and Human Affairs Program of the Battelle Memorial Institute of Geneva between 1972 and 1976, was used for researching and solving the complicated and intertwined problem group (Fontela and Gabus 1976; Gabus and Fontela1972; Warfield1976). DEMATEL was developed in the belief that pioneering and appropriate use of scientific research methods could improve understanding of the spe-cific problematic, the cluster of intertwined problems, and contribute to identification of workable solutions by a hierarchical structure. The methodology, according to the concrete characteristics of objective affairs, can confirm the interdependence among the variables/attributes and restrict the relation that reflects the characteristics with an essential system and development trend (Hori and Shimizu 1999; Tzeng et al.2007). Using the DEMATEL method to size and process individual subjective perceptions, brief and impressionistic human insights into problem complexity can be gained. Following the DEMATEL process the end product of the analysis is a visual representation, an indi-vidual map of the mind, according to which the respondent organizes his own action in the world, if he is to keep internally coherent to respect his implicit priorities and to reach his secret goals.
The steps of the DEMATEL method can be described as follows:
2.1 Step 1: calculate the average matrix by scores Respondents are asked to indicate the direct influence that they believe each factor/element i exerts on each factor/element j of the others, as indicated by aij, using an
integer scale ranging, for example from 0 to 4 (going from ‘‘No influence (0),’’ to ‘‘Extreme strong influence (4)’’). The higher score indicates that the respondent has expressed that the insufficient involvement in problem of factor i exerts the stronger possible direct influence on the inability of factor j, or, in positive terms, that greater improvement i is required to improve j.
From any group of direct matrices of respondents it is possible to derive an average matrix A. Each element of this average matrix will be in this case the mean of the same elements in the different direct matrices of the respondents.
2.2 Step 2: calculate the initial direct influence matrix The initial direct influence matrix D can be obtained by normalizing the average matrix A, in which all principal diagonal elements are equal to zero. Based on matrix D, the initial influence which a factor dispatches to and receives from another is shown.
The elements of matrix D portrays a contextual relation among the elements of the system and can be converted into a visible structural model, an impact- digraph-map, of the system with respect to that relation. For example, as in Fig.1, the respondents are asked to indicate only direct links. In the directed digraph graph represented here, factor i affects directly only factors j and k; indirectly, it also affects first l, m and n and, secondly, o and q.
2.3 Step 3: derive the full direct/indirect influence matrix
A continuous decrease of the indirect effects of problems along the powers of matrix D, e.g., D2, D3 ,…, D?, and therefore guarantees convergent solutions to matrix inver-sion. In a configuration such as Fig.1, the influence exerted by factor i on factor p will be smaller than the one that it exerts on factor m, and again smaller than the one exerted
i k n q m o l j
on factor j. This being so, the infinite series of direct and indirect effects can be illustrated. Let the (i, j) element of matrix A is denoted by aij, the matrix can be gained
following Eqs.1–4. D¼ s A; s[ 0 ð1Þ or ½di jnn¼ s ½ai jnn; s[ 0; i; j2 f1; 2; . . .; ng ð2Þ where s¼ Min 1 max1 i nPnj¼1 aij ; 1 max1 i nPni¼1 aij " # ð3Þ and lim m!1D m¼ ½0 nn; where D¼ dij n x n; 0 dij\1 ð4Þ where 0 Pnj¼1dijor Pni¼1dij n o
\1 and at least one Pn
j¼1dijor P n
i¼1dij equal 1, 8i; j 2 f1; 2; . . .; ng; then
lim
m!1D
m¼ ½0 nn:
The full direct/indirect influence matrix F, the infinite series of direct and indirect effects of each factor, can be obtained by the matrix operation of D. The matrix F can show the final structure of factors after the continuous process (see Eq.5). Let Wi(f) denote the normalized ith
row sum of matrix F, thus, the Wi(f) value means the sum
of influence dispatching from factor i to the other factors both directly and indirectly. The Vi(f), the normalized ith
column sum of matrix F, means the sum of influence that factor i receives from the other factors.
The total-influence matrix T can be obtained by using Eq.5where I is denoted as the identity matrix.
F¼ D þ D2þ þ Dm ¼X m i¼1 Di¼ D I Dð Þ1; when m! 1 ð5Þ
2.4 Step 4: set threshold value and obtain the impact-digraph- map
Setting a threshold value, p, to filter the obvious effects denoted by the elements of matrix F, is necessary to explain the structure of factors. Base on the matrix F, each element, fij, of matrix F provides the information about a
factor i dispatches influence to factor j, or, in another word, factor j receives influence from factor i. If all the infor-mation from matrix F converts to the impact-digraph-map, it will be too complex to show the necessary information for decision-making. In order to obtain an appropriate impact-digraph-map, setting a threshold value of the
influence level is necessary for the decision maker. Only some elements, whose influence level in matrix F higher than the threshold value, can be chosen and converted into the impact-digraph-map.
The threshold value is decided by the decision makers or, in this paper, experts through discussions. Like matrix D, contextual relation among the elements of matrix F can also be converted into a digraph map. If the threshold value is too low, the map will be too complex to show the necessary information for decision-making. If the threshold value is too high, many factors will be presented as inde-pendent factors without relations to another factor. Each time the threshold value increases, some factors or rela-tionship will be removed from the map. An appropriate threshold value is necessary to obtain a suitable impact-digraph-map and proper information for the further anal-ysis and decision-making.
After threshold value and relative impact-digraph-map are decided, the final influence result can be shown. For example, the impact-digraph-map of a factor is the same as Fig.1 and eight elements exist in this map. Because of continuous direct/indirect effects between them, finally, the effectiveness of these eight elements could be considered to be represented by two independent final affected elements: o and q. The other components, not shown in the impact-digraph-map, of a factor can be considered as independent elements because no obvious interrelation with others exists.
3 Fuzzy decision maps
In order to deal with the problem of dependence and feedback among criteria, we first depict the FCM as shown in Fig.2 to illustrate the situation of decision making. In Fig.2, eijdenotes the interaction effect from the ith
crite-rion to the jth critecrite-rion, and eii indicates the compound
effect of the ith criterion by self-relation. As we know, due to the problem of compound and interaction effects, it is hard for decision makers to make a good decision using the simple weighted method.
One way to overcome the problems above is to obtain the information of influences among criteria and then to derive the finial weights by considering the influences among criteria. However, since these criteria may have loop or feedback relationships, it is hard to derive the influences among criteria. Next, we first employ the FCM to derive the influence among criteria and then obtain the finial weights by using the weighted formulation.
FCM, which was first proposed by Koska (1988) and Sekitani and Takahashi (2001), extends the original cog-nitive maps of political elite (Axelrod 1976) by incorpo-rating fuzzy measures to provide a flexible and realistic method for extracting the fuzzy relationships among objects
in a complex system. Therefore, recently, FCM have been widely employed in the applications of political decision-making, business management, industrial analysis, and system control (Andreou et al. 2005; Pagageorgiou and Groumpos2005; Stylios and Groumpos2004), except for the area of MCDM. The concepts of FCM can be described as follows.
Given a 4-tuple (N, E, C, f) where N = {N1, N2,…, Nn}
denotes the set of n objects, E denotes the connection matrix (initial direct-influence matrix) which is composed of the weights between objects, C is the state matrix, where C(0) is the initial matrix and C(t) is the transition-state matrix at certain iteration t, and f is a threshold function, which indicates the weighting relationship between C(t)and C(t?1). Several formulas have been used as threshold functions such as
fðxÞ ¼ 1 if x 1
0 if x\1; ðHard line functionÞ
fðxÞ ¼ tanhðxÞ ¼ ð1 exÞ=ð1 þ exÞ; Hyperbolic tangent function
ð Þ
and
fðxÞ ¼ 1=ð1 þ exÞ: ðLogistic functionÞ
The different threshold functions will get the discriminative numerical results and a sensitive analysis was proposed by Chen et al. (2008).
The influence of the specific criterion to other criteria can be calculated using the following updating equation:
Cðtþ1Þ¼ f ðCðtÞEÞ; Cð0Þ¼ Inn ð6Þ
where In9ndenotes the identity matrix.
The vector–matrix multiplication operation to derive successive FCM transition-states is iterated until it con-verges to a fixed point situation or a limit steady-state cycle. The steady-state vector–matrix remains unchanged for successive iterations is called a fixed point situation and the sequence of the steady-state vector–matrix keeps repeating indefinitely is called a limit state cycle. Now, we
can summarize the proposed method to derive the priorities of criteria as follows:
Step 1 Compare the importance among criteria to derive the local weight vector using the eigenvalue approach;
Step 2 Depict the fuzzy cognitive map (FCM) to build the influence among criteria by the expert; Step 3 Calculate Eq.6 for obtaining the steady-state
matrix;
Step 4 Derive the global weight vector. In order to derive the global weights, we should first normalize the local weight vector (z) and the steady-state matrix (C*) as follows: zt¼ 1 kz; ð7Þ and Ct ¼1 cC ð8Þ
where k is the largest element of z and c is the largest row sum of C*. Then, we can obtain the global weight vector by using the following weighting equation:
w¼ ztþ Ctzt: ð9Þ
3.1 Mathematical proof
In this section, we show that DEMATEL is the specific case of FDM when the threshold function is linear. Both FDM and DEMATEL have the same direct and indirect influence matrix.
Consider the following threshold function using in FDM.
fðxÞ ¼ x ðpure linear functionÞ
In FDM, it uses Eq.6 to obtain the direct and indirect influence matrix.
C0¼ Inn; C1¼ C0E¼ E; E is the initial
direct-rela-tion matrix. Ct¼f ðCt1EþC0Þ
¼ðCt1EþC0ÞE
¼f ðCt2ÞE2þE
¼ððCt2EþC0ÞE2þE¼f ðCt3ÞE3þE2þE
¼ððCt3EþC0ÞE3þE2þE¼f ðCt4ÞE4þE3þE2þE
...
¼EtþEt1þþE
¼EðIþEþE2þþEt1ÞðIEÞðIEÞ1
¼E½ðIþEþE2þþEt1ÞðIEÞðIEÞ1
¼E½ðIþEþE2þþEt1ÞðEþE2þþEtÞðIEÞ1
¼EðIEtÞðIEÞ1 Criterion 1 Criterion 4 Criterion 5 Criterion 3 Criterion 2 e21 e 31 e 14 e 25 e 52 e 55 e 33 e 34
and lim t!1E t¼ ½0 nn; where E¼ ½eijnn; 0 eij\1 and Xn i¼1 eij 1: Therefore, Ct¼ lim t!1 Pt i¼1E i¼ EðI EÞ1 ; then the formula of FDM equals to that of DEMETEL (Eq.5). So, DEMATEL is a specific case of FDM.
Next it uses two numerical examples to demonstrate the proposed results in Sect.4 and two extra examples in Appendix3to support our research findings.
4 Numerical examples
In this section, two numerical examples are illustrated to demonstrate the proposed results and compared the results between in the FDM and the DEMATEL. The first example is the multi-criteria decision problem about suppliers-eval-uation. Then the second example is the more complicated decision problem about customers’ evaluation to product purchasing. Note that in this paper we use the threshold function that is the pure-linear function in FDM to indicate the relationships among criteria. In Appendix3, two extra examples are proposed to reinforce our research results. Example 1 Consider a decision maker tries to select the suppliers according the following criteria including Quality (Q), Matching (M), Lead-time (L), and Costs (C). For choosing the best alternative, we should derive the influ-ence scores of each criterion and calculate the influinflu-ence scores of each supplier. Figure3and scoring matrix show the relationships between criteria (scale: 0 no influences; 1 low influences; 2 moderate influences; 3 strong influences; 4 extreme strong influences).
And scoring matrix as
S¼ Q C L M Q C L M 0 4 1 0 2 0 0 3 1 2 0 0 4 2 2 0 2 6 6 6 4 3 7 7 7 5 ; max j Pn i¼1eij¼ max j ½7; 8; 3; 3 ¼ 8 and max i Pn j¼1 eij¼ max j ½5; 5; 3; 8 k¼ min i;j 1 max j Pn i¼1 eij ; 1 max i Pn j¼1 eij 2 6 6 6 4 3 7 7 7 5¼ 1 8:
From the Fig.3and scoring matrix, we can calculate the grades/degrees of scoring matrix by dividing the max row
sum, i.e., E = kS. The grades/degrees of direct influence matrix as follow: E¼ Q C L M Q C L M 0 0:50 0:125 0 0:25 0 0 0:375 0:125 0:25 0 0 0:5 0:25 0:25 0 2 6 6 6 4 3 7 7 7 5
Next, we can obtain the steady-state matrix by calculating Eq.6 in the FDM method and Eq.5 in the DEMATEL as follows (see Appendix1):
DEMATEL Quality Costs Lead-time Matching Quality 0.4005 0.8428 0.2541 0.3160 Costs 0.7126 0.5616 0.2355 0.5856 Lead-time 0.3532 0.4957 0.0906 0.1859 Matching 0.9667 0.9357 0.4586 0.3509 FDM (f(x) = x) Quality 0.4004a 0.8427a 0.2541 0.3160 Costs 0.7126 0.5615a 0.2355 0.5856 Lead-time 0.3532 0.4957 0.0906 0.1859 Matching 0.9667 0.9357 0.4585a 0.3509 a Means the difference between FDM and DEMATEL is 0.0001
Example 2 In this example, consider a customer to pur-chase a product according to the following five criteria including Quality (Q), Delivery (D), Price (P), Yield (Y), and Service (S). Figure4 and scoring matrix display the relationships between criteria (Scale: 0 no influences; 1 low influences; 2 moderate influences; 3 strong influences; 4 extreme strong influences).
Quality Lead-time Matching Costs 4 2 2 1 1 3 2 4 2
And scoring matrix as E¼ Q D P Y S Q D P Y S 0 2 4 1 2 0 0 1 0 0 2 3 0 0 4 0 0 0 0 2 0 0 2 3 0 2 6 6 6 6 6 4 3 7 7 7 7 7 5 ; max j Xn i¼1 eij¼ max j ½2; 5; 7; 4; 8 ¼ 8 and max i Pn j¼1 eij¼ max j ½9; 1; 9; 2; 5 ¼ 9; then k¼ min i;j 1 max j Pn i¼1 eij ; 1 max i Pn j¼1 eij 2 6 6 6 4 3 7 7 7 5¼ 1 9:
From the Fig.4and scoring matrix, we can calculate the grades/degrees of scoring matrix by dividing the max row sum, i.e., E = kS. The grades/degrees of direct influence matrix as follow: E¼ Q D P Y S Q D P Y S 0 0:2222 0:4444 0:1111 0:2222 0 0 0:1111 0 0 0:2222 0:3333 0 0 0:4444 0 0 0 0 0:2222 0 0 0:2222 0:3333 0 2 6 6 6 6 6 4 3 7 7 7 7 7 5
Next, we can obtain the steady-state matrix by calculating Eq.6 in the FDM method and Eq.5 in the DEMATEL as follows (see Appendix2):
Quality Delivery Price Yield Service DEMATEL Quality 0.1589 0.4958 0.7150 0.3461 0.6522 Delivery 0.0334 0.0575 0.1504 0.0307 0.0811 Price 0.3007 0.5179 0.3533 0.2766 0.7297 Yield 0.0160 0.0276 0.0722 0.0947 0.2789 Service 0.0722 0.1243 0.3248 0.4263 0.2551 FDM (f(x) = x) Quality 0.1589 0.4958 0.7150 0.3461 0.6521a Delivery 0.0334 0.0575 0.1503a 0.0307 0.0811 Price 0.3007 0.5179 0.3533 0.2766 0.7297 Yield 0.0160 0.0276 0.0722 0.0947 0.2789 Service 0.0722 0.1243 0.3247a 0.4263 0.2551 a Means the difference between FDM and DEMATEL is 0.0001
Both calculation processes of the numerical examples were collected in Appendix1and2. Two extra examples in
Appendix3are proposed to support our research findings. Next section, we provide the depth discussions according to the results of the numerical examples above in Sect.5.
5 Discussions
Structural MADM problems involve determining the best options by considering the effects among criteria. In order to solve the problems above, the major point is to derive the direct and indirect influence matrix. After getting a direct and indirect influence matrix, it can obtain the impact diagraph map. Although the DEMATEL has been proposed to deal this problem, but it has the limitation that the problems influence must be interactive linearly.
In this paper, the FDMs, which combine the eigenvalue method, the FCMs, and the weighting equation, are pro-posed to deal with the structural MADM problems. The direct and indirect influence matrix is derived by using the updating equation. Finally, it reaches the steady-state matrix. Under the threshold function is linear, both the DEMATEL and FDM have the direct/indirect influence matrix.
From the results of two numerical examples and math-ematical proof above, it exposes three main shortcomings of the DEMATEL method. First, the criteria states may not be interactive linearly in some real world situations. Sec-ond, the initial direct matrix of DEMATEL method has the characteristic that its all principal diagonal elements are equal to zero. This characteristic means the DEMATEL method can not handle the criterion which contains self-feedback loop shown in Fig.2. Third, both the DEMATEL method and FDM method have the discrete-time process
Quality Yield Service Delivery Price 2 1 2 4 2 4 2 3 2 1 3
with the Markov property. The DEMATEL processes the criteria in different states with dynamical linearly.
In contrast, the benefit of the FDM methods can be summarized as follows. First, the FDM method modifies the shortcoming of the DEMATEL. We can employ the different threshold functions to indicate the various kinds of relationship among criteria. The more flexible threshold functions may have better fitness when construct the direct and indirect influence matrix. Second, the FDM method can treat the problem of compound and interaction effect of the criterion containing self feedback loop. Third, In the FDM method, it can process the dynamic criteria status and exceeds the limitation of the DEMATEL method with dynamical non-linearly.
6 Conclusions
The MCDM problems with the complicated intertwined problem group are hard for the decision-maker to make a good decision. Although the DEMATEL method has been widely used to deal with this problem, the shortcoming should be overcome for proving the satisfaction solution. In this paper, the FDM method is proposed to deal with the structural MCDM problems. Without limiting by the problems influence must be interactive linearly, it can employ the flexible threshold function to get proper solutions in real world cases. On the basis of the numerical results and mathematical proof, we can con-clude that FDM is a generalization of DEMATEL method.
Appendix 1
Result of the supplier evaluation in DEMATAL for the operation processes of Example 1
I¼ 1:00 0:00 0:00 0:00 0:00 1:00 0:00 0:00 0:00 0:00 1:00 0:00 0:00 0:00 0:00 1:00 2 6 6 4 3 7 7 5 D¼ 0:000 0:500 0:125 0:000 0:250 0:000 0:000 0:375 0:125 0:250 0:000 0:000 0:500 0:250 0:250 0:000 2 6 6 4 3 7 7 5 I D ¼ 1:000 0:500 0:125 0:000 0:250 1:000 0:000 0:375 0:125 0:250 1:000 0:000 0:500 0:250 0:250 1:000 2 6 6 6 4 3 7 7 7 5 ðI DÞ1¼ 1:4005 0:8428 0:2541 0:3160 0:7126 1:5616 0:2355 0:5856 0:3532 0:4957 1:0906 0:1859 0:9667 0:9357 0:4586 1:3509 2 6 6 4 3 7 7 5 F¼ DðI DÞ1¼ 0:4005 0:8428 0:2541 0:3160 0:7126 0:5616 0:2355 0:5856 0:3532 0:4957 0:0906 0:1859 0:9667 0:9357 0:4586 0:3509 2 6 6 4 3 7 7 5
Result of the supplier evaluation in FDM for the operation processes of Example 1
E¼ D E¼ 0:0000 0:5000 0:1250 0:0000 0:2500 0:0000 0:0000 0:3750 0:1250 0:2500 0:0000 0:0000 0:5000 0:2500 0:2500 0:0000 2 6 6 4 3 7 7 5 C0¼ I C0¼ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 2 6 6 4 3 7 7 5 C1¼ C0E¼ 0:0000 0:5000 0:1250 0:0000 0:2500 0:0000 0:0000 0:3750 0:1250 0:2500 0:0000 0:0000 0:5000 0:2500 0:2500 0:0000 2 6 6 4 3 7 7 5 C2¼ ðC1þ C0Þ E ¼ 0:1406 0:5313 0:1250 0:1875 0:4375 0:2188 0:1250 0:3750 0:1875 0:3125 0:0156 0:0938 0:5938 0:5625 0:3125 0:0938 2 6 6 4 3 7 7 5 C3¼ ðC2þ C0Þ E ¼ 0:2422 0:6484 0:1895 0:1992 0:5078 0:3438 0:1484 0:4570 0:2520 0:3711 0:0469 0:1172 0:7266 0:6484 0:3477 0:2109 2 6 6 4 3 7 7 5 … C24¼ ðC23þ C0Þ E ¼ 0:4004 0:8427 0:2541 0:3160 0:7126 0:5615 0:2355 0:5856 0:3532 0:4957 0:0906 0:1859 0:9666 0:9357 0:4585 0:3509 2 6 6 4 3 7 7 5 C25¼ ðC24þ C0Þ E ¼ 0:4004 0:8427 0:2541 0:3160 0:7126 0:5615 0:2355 0:5856 0:3532 0:4957 0:0906 0:1859 0:9667 0:9357 0:4585 0:3509 2 6 6 4 3 7 7 5
Appendix 2
Result of the customer evaluation in DEMATAL for the operation processes of Example 2
I¼ 1:00 0:00 0:00 0:00 0:00 0:00 1:00 0:00 0:00 0:00 0:00 0:00 1:00 0:00 0:00 0:00 0:00 0:00 1:00 0:00 0:00 0:00 0:00 0:00 1:00 2 6 6 6 6 4 3 7 7 7 7 5 D¼ 0:0000 0:2222 0:4444 0:1111 0:2222 0:0000 0:0000 0:1111 0:0000 0:0000 0:2222 0:3333 0:0000 0:0000 0:4444 0:0000 0:0000 0:0000 0:0000 0:2222 0:0000 0:0000 0:2222 0:3333 0:0000 2 6 6 6 6 4 3 7 7 7 7 5 I D ¼ 1:0000 0:2222 0:4444 0:1111 0:2222 0:0000 1:0000 0:1111 0:0000 0:0000 0:2222 0:3333 1:0000 0:0000 0:4444 0:0000 0:0000 0:0000 1:0000 0:2222 0:0000 0:0000 0:2222 0:3333 1:0000 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 ðI DÞ1¼ 1:1589 0:4958 0:7150 0:3461 0:6522 0:0334 1:0575 0:1504 0:0307 0:0811 0:3007 0:5179 1:3533 0:2766 0:7297 0:0160 0:0276 0:0722 1:0947 0:2789 0:0722 0:1243 0:3248 0:4263 1:2551 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5 F¼ DðI DÞ1 ¼ 0:1589 0:4958 0:7150 0:3461 0:6522 0:0334 0:0575 0:1504 0:0307 0:0811 0:3007 0:5179 0:3533 0:2766 0:7297 0:0160 0:0276 0:0722 0:0947 0:2789 0:0722 0:1243 0:3248 0:4263 0:2551 2 6 6 6 6 6 4 3 7 7 7 7 7 5
Result of the customer evaluation in FDM for the operation processes of Example 2
E¼ D E¼ 0 0:2222 0:4444 0:1111 0:2222 0 0 0:1111 0 0 0:2222 0:3333 0 0 0:4444 0 0 0 0 0:2222 0 0 0:2222 0:3333 0 2 6 6 6 6 4 3 7 7 7 7 5 C0¼ I C0¼ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 2 6 6 6 6 4 3 7 7 7 7 5 C1¼ C0 E ¼ 0 0:2222 0:4444 0:1111 0:2222 0 0 0:1111 0 0 0:2222 0:3333 0 0 0:4444 0 0 0 0 0:2222 0 0 0:2222 0:3333 0 2 6 6 6 6 4 3 7 7 7 7 5 C2¼ ðC1þ C0Þ E ¼ 0:0987 0:3703 0:5185 0:1852 0:4444 0:0247 0:0370 0:1111 0 0:0494 0:2222 0:3827 0:2345 0:1728 0:4938 0 0 0:0494 0:0741 0:2222 0:0494 0:0741 0:2222 0:3333 0:1728 2 6 6 6 6 4 3 7 7 7 7 5 C3¼ ðC2þ C0Þ E ¼ 0:1152 0:4169 0:6282 0:2702 0:5157 0:0247 0:0425 0:1372 0:0192 0:0549 0:2743 0:4608 0:2510 0:1893 0:6364 0:0110 0:0165 0:0494 0:0741 0:2606 0:0494 0:0850 0:2908 0:3964 0:1838 2 6 6 6 6 4 3 7 7 7 7 5 … C17¼ ðC16þ C0Þ E ¼ 0:1589 0:4958 0:7150 0:3461 0:6521 0:0334 0:0575 0:1503 0:0307 0:0811 0:3007 0:5179 0:3533 0:2766 0:7297 0:0160 0:0276 0:0722 0:0947 0:2789 0:0722 0:1243 0:3247 0:4263 0:2551 2 6 6 6 6 4 3 7 7 7 7 5 C18¼ ðC17þ C0Þ E ¼ 0:1589 0:4958 0:7150 0:3461 0:6521 0:0334 0:0575 0:1503 0:0307 0:0811 0:3007 0:5179 0:3533 0:2766 0:7297 0:0160 0:0276 0:0722 0:0947 0:2789 0:0722 0:1243 0:3247 0:4263 0:2551 2 6 6 6 6 4 3 7 7 7 7 5 Appendix 3
Two extra examples are proposed to reinforce our research results. Example 3 shows how to evaluate the human capital. Example 4 shows how to evaluate the external structure capital.
Example 3 The human capital includes four indices: leadership (LS), turnover of professional employees (TPE), replacement cost of professional employees (RPE), and team work (TW). The grades/degrees of direct influence matrix as follow: E¼ LS TPE RPE TW LS TPE RPE TW 0 0:3462 0:1923 0:4615 0:1923 0 0:1538 0:1154 0:1538 0:3462 0 0:1538 0:2692 0:3846 0:1923 0 2 6 6 6 4 3 7 7 7 5
Next, we can obtain the steady-state matrix by calculating Eq.6 in the FDM method and Eq.5 in the DEMATEL as follows.
DEMATEL LS TPE RPE TW
Leadership (LS) 0.5844 1.1515 0.6677 0.9668 Turnover of professional employees (TPE) 0.4648 0.4714 0.4015 0.4460 Replacement cost of professional employees (RPE) 0.5128 0.8462 0.3325 0.5393 Team work (TW) 0.7039 1.0386 0.5904 0.5355 FDM (f(x) = x) LS TPE RPE TW Leadership (LS) 0.5843a 1.1514a 0.6677 0.9667a TPE 0.4648 0.4714 0.4014a 0.4460 RPE 0.5128 0.8462 0.3325 0.5392a TW 0.7039 1.0386 0.5904 0.5355
a Means the difference between FDM and DEMATEL is 0.0001
In above table, the numerical results show the DEMA-TEL method and the FDM method using linear function almost the same. This finding supports the FDM is a gen-eral methods of the DEMATEL method.
Example 4 The external structure capital includes five indices: market share (MS), customer satisfaction (CS), market growth rate (MGR), brand loyalty (BL), and future prospective of product market (FP). The grades/degrees of direct influence matrix as follow:
E¼ MS CS MGR BL FP MS CS MGR BL FP 0 0:2903 0:1935 0:2581 0:1613 0:3226 0 0:2258 0:3226 0:1290 0:1613 0:0645 0 0:0645 0:2903 0:3226 0:2903 0:1290 0 0:0968 0:0968 0:0323 0:3548 0:0323 0 2 6 6 6 6 6 6 4 3 7 7 7 7 7 7 5
Next, we can obtain the steady-state matrix by calculating Eq.6 in the FDM method and Eq.5 in the DEMATEL as follows.
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DEMATEL MS CS MGR BL FP
Market share (MS) 0.7212 0.8059 0.8677 0.7831 0.7093 Customer satisfaction (CS) 1.0403 0.6438 0.9494 0.8840 0.7410 Market growth rate (MGR) 0.5287 0.3725 0.4425 0.3687 0.5878 Brand loyalty (BL) 0.9660 0.8133 0.8048 0.5845 0.6477 Future prospective of product market (FP) 0.4190 0.2895 0.6525 0.2863 0.3221 continued FDM (f(x) = x) MS CS MGR BL FP MS 0.7212 0.8059 0.8677 0.7831 0.7093 CS 1.0403 0.6438 0.9494 0.8840 0.7410 MGR 0.5287 0.3725 0.4425 0.3687 0.5878 BL 0.9660 0.8133 0.8048 0.5845 0.6477 Future prospective of FP 0.4190 0.2895 0.6525 0.2863 0.3221
In above table, the numerical results are the same in both two methods. The DEMATEL method and the FDM (f(x) = x) have the same values
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