Fuzzy decision maps: a generalization of the DEMATEL methods

Soft Comput (2010) 14:1141–1150 DOI 10.1007/s00500-009-0507-0 ORIGINAL PAPER 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 analyze 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. 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 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 Hwang 1992). 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 (Fishburn 1970), 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 independence or utility separability seems to be irrational. Therefore, it is interesting to clarify the structure among criteria, and then we can determine the appropriate MADM methods 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 123 1142 Tzeng 2006) for solving MCDM problems in preferential weights by using the AHP (independence) or ANP (dependence and feedback) or fuzzy integral (inter-dependent/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 mathematical 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. 5 and 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 Fontela 1972; Warfield 1976). DEMATEL was developed in the belief that pioneering and appropriate use of scientific research methods could improve understanding of the specific 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 individual 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: G.-H. Tzeng et al. 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 inversion. 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 j n m l 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 123 q Fig. 1 An example of direct graph o Fuzzy decision maps 1143 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. D ¼ s  A; ð1Þ s[0 or ½di j nn ¼ s  ½ai j nn ; s [ 0; i; j 2 f1; 2; . . .; ng ð2Þ where s ¼ Min " 1 Pn max1  i  n j¼1 1 Pn ; aij max1  i  n i¼1 aij # ð3Þ and lim Dm ¼ ½0nn ; m!1 nP   where D ¼ dij n x n ; Pn 0  dij \1 ð4Þ o n where 0  j¼1 dij or i¼1 dij \1 and at least one Pn Pn j¼1 dij or i¼1 dij equal 1, 8i; j 2 f1; 2; . . .; ng; then lim Dm ¼ ½0nn : m!1 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. 5 where I is denoted as the identity matrix. F ¼ D þ D2 þ    þ Dm ¼ m X Di ¼ DðI  DÞ1 ; i¼1 when m ! 1 ð5Þ 2.4 Step 4: set threshold value and obtain the impactdigraph- 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 information 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 independent factors without relations to another factor. Each time the threshold value increases, some factors or relationship will be removed from the map. An appropriate threshold value is necessary to obtain a suitable impactdigraph-map and proper information for the further analysis 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 impactdigraph-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, eij denotes the interaction effect from the ith criterion to the jth criterion, 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 cognitive maps of political elite (Axelrod 1976) by incorporating fuzzy measures to provide a flexible and realistic method for extracting the fuzzy relationships among objects 123 1144 G.-H. Tzeng et al. Criterion 1 e21 Criterion 2 can summarize the proposed method to derive the priorities of criteria as follows: e 55 e 14 Step 1 e 52 Criterion 5 e 25 Step 2 e 33 e 31 Step 3 Criterion 3 Criterion 4 Step 4 e 34 Fig. 2 The problem of a decision map in a complex system. Therefore, recently, FCM have been widely employed in the applications of political decisionmaking, business management, industrial analysis, and system control (Andreou et al. 2005; Pagageorgiou and Groumpos 2005; Stylios and Groumpos 2004), 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  1 if x  1 ; ðHard line functionÞ f ðxÞ ¼ 0 if x\1 f ðxÞ ¼ tanhðxÞ ¼ ð1  ex Þ=ð1 þ ex Þ; ðHyperbolic  tangent functionÞ 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: Cðtþ1Þ ¼ f ðCðtÞ EÞ; Cð0Þ ¼ I nn ð6Þ where In9n denotes the identity matrix. The vector–matrix multiplication operation to derive successive FCM transition-states is iterated until it converges 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 123 1 z; k ð7Þ and Ct ¼ 1  C c 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: w ¼ zt þ Ct zt : 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. f ðxÞ ¼ x and f ðxÞ ¼ 1=ð1 þ ex Þ: zt ¼ Compare the importance among criteria to derive the local weight vector using the eigenvalue approach; Depict the fuzzy cognitive map (FCM) to build the influence among criteria by the expert; Calculate Eq. 6 for obtaining the steady-state matrix; 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: ðpure  linear functionÞ In FDM, it uses Eq. 6 to obtain the direct and indirect influence matrix. C0 ¼ I nn ; C1 ¼ C0 E ¼ E; E is the initial direct-relation matrix. Ct ¼f ðCt1 EþC0 Þ ¼ðCt1 EþC0 ÞE ¼f ðCt2 ÞE2 þE ¼ððCt2 EþC0 ÞE2 þE¼f ðCt3 ÞE3 þE2 þE ¼ððCt3 EþC0 ÞE3 þE2 þE¼f ðCt4 ÞE4 þE3 þE2 þE ... ¼Et þEt1 þþE ¼EðIþEþE2 þþEt1 ÞðIEÞðIEÞ1 ¼E½ðIþEþE2 þþEt1 ÞðIEÞðIEÞ1 ¼E½ðIþEþE2 þþEt1 ÞðEþE2 þþEt ÞðIEÞ1 ¼EðIEt ÞðIEÞ1 Fuzzy decision maps 1145 and lim Et ¼ ½0nn ; where E ¼ ½eij nn ; Quality t!1 n X 0  eij \1 and 4 eij  1: i¼1 4 Therefore, C ¼ lim i¼1 Ei ¼ EðI  EÞ1 ; then the t!1 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 Appendix 3 to support our research findings. Pt t 1 2 3 Matching Costs 2 1 2 2 Lead-time 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-evaluation. 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 Appendix 3, 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 influence scores of each criterion and calculate the influence scores of each supplier. Figure 3 and 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 Q C L M 3 2 Q 0 4 1 0 P 7 6 S ¼ C 6 2 0 0 3 7 ; max ni¼1 eij ¼ max 7 6 j j L 41 2 0 05 M 4 2 2 0 n P ½7; 8; 3; 3 ¼ 8 and max eij ¼ max ½5; 5; 3; 8 i 3 6 7 1 1 1 6 7 k ¼ min6 ; 7¼ : n n P P i;j 4 5 8 max eij max eij j i¼1 i sum, i.e., E = kS. The grades/degrees of direct influence matrix as follow: Q 0 6 E ¼ C 6 0:25 6 L 4 0:125 0:5 M Q 2 C L 0:50 0:125 0 0 0:25 0 0:25 0:25 M 3 0 0:375 7 7 7 0 5 0 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 Appendix 1): 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 Quality 0.4004a 0.8427a 0.2541 0.3160 Costs 0.7126 0.5615a 0.2355 0.5856 FDM (f(x) = x) a Lead-time 0.3532 0.4957 0.0906 0.1859 Matching 0.9667 0.9357 0.4585a 0.3509 Means the difference between FDM and DEMATEL is 0.0001 j j¼1 2 Fig. 3 The evaluation of a supplier j¼1 From the Fig. 3 and scoring matrix, we can calculate the grades/degrees of scoring matrix by dividing the max row Example 2 In this example, consider a customer to purchase a product according to the following five criteria including Quality (Q), Delivery (D), Price (P), Yield (Y), and Service (S). Figure 4 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). 123 1146 G.-H. Tzeng et al. And scoring matrix as 3 Q D P Y S Q 20 2 4 1 23 7 D6 60 0 1 0 07 7; 6 E¼ P 62 3 0 0 47 7 6 Y 40 0 0 0 25 S 0 0 2 3 0 n X max eij ¼ max ½2; 5; 7; 4; 8 ¼ 8 j Delivery 2 2 4 j i¼1 j i¼1 i 3 2 Service Yield 2 Fig. 4 The product evaluation of a customer j¼1 From the Fig. 4 and 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: Q Q2 0 D6 6 0 6 E¼ P 6 0:2222 6 Y 4 0 0 S 4 1 7 6 1 1 7 1 6 ¼ : k ¼ min6 ; n n P P 7 i;j 4 5 9 max eij max eij D P 0:2222 0:4444 0 0:1111 0:3333 0 0 0 0 0:2222 Y S 3 0:1111 0:2222 0 0 7 7 7 0 0:4444 7 7 0 0:2222 5 0:3333 0 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 Appendix 2): Quality Delivery Price Yield Service 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 0.3461 0.6521a DEMATEL FDM (f(x) = x) Quality 0.1589 0.4958 0.7150 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 Means the difference between FDM and DEMATEL is 0.0001 Both calculation processes of the numerical examples were collected in Appendix 1 and 2. Two extra examples in 123 2 Quality and n P eij ¼ max ½9; 1; 9; 2; 5 ¼ 9; then max i j j¼1 3 2 a Price 1 Appendix 3 are 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 proposed 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 mathematical 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. Second, 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 selffeedback loop shown in Fig. 2. Third, both the DEMATEL method and FDM method have the discrete-time process Fuzzy decision maps 1147 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 conclude that FDM is a generalization of DEMATEL method. Appendix 1 Result of the supplier evaluation in DEMATAL for the operation processes of Example 1 2 3 1:00 0:00 0:00 0:00 6 0:00 1:00 0:00 0:00 7 7 I¼6 4 0:00 0:00 1:00 0:00 5 0:00 0:00 0:00 1:00 2 3 0:000 0:500 0:125 0:000 6 0:250 0:000 0:000 0:375 7 7 D¼6 4 0:125 0:250 0:000 0:000 5 0:500 0:250 0:250 0:000 2 1:000 0:500 0:125 6 0:250 1:000 0:000 6 ID¼6 4 0:125 0:250 1:000 0:500 0:250 0:250 3 0:000 0:375 7 7 7 0:000 5 1:000 2 3 1:4005 0:8428 0:2541 0:3160 6 0:7126 1:5616 0:2355 0:5856 7 7 ðI  DÞ1 ¼ 6 4 0:3532 0:4957 1:0906 0:1859 5 0:9667 0:9357 0:4586 1:3509 2 3 0:4005 0:8428 0:2541 0:3160 6 0:7126 0:5616 0:2355 0:5856 7 7 F ¼ DðI  DÞ1 ¼ 6 4 0:3532 0:4957 0:0906 0:1859 5 0:9667 0:9357 0:4586 0:3509 Result of the supplier evaluation in FDM for the operation processes of Example 1 E¼D 2 0:0000 6 0:2500 E¼6 4 0:1250 0:5000 C0 ¼ I 2 0:5000 0:0000 0:2500 0:2500 3 0 0 0 07 7 1 05 0 1 2 0:0000 6 0:2500 1 0 C ¼C E¼6 4 0:1250 0:5000 1 6 0 C0 ¼ 6 40 0 0:1250 0:0000 0:0000 0:2500 3 0:0000 0:3750 7 7 0:0000 5 0:0000 0:5000 0:0000 0:2500 0:2500 0:1250 0:0000 0:0000 0:2500 0 1 0 0 1 0 C2 ¼ ðC 2 þC ÞE 0:1406 0:5313 6 0:4375 0:2188 ¼6 4 0:1875 0:3125 0:5938 0:5625 2 0 C3 ¼ ðC 2 þC ÞE 0:2422 0:6484 6 0:5078 0:3438 ¼6 4 0:2520 0:3711 0:7266 0:6484 0:1250 0:1250 0:0156 0:3125 3 0:1875 0:3750 7 7 0:0938 5 0:0938 0:1895 0:1484 0:0469 0:3477 3 0:1992 0:4570 7 7 0:1172 5 0:2109 … 23 0 C24 ¼ ðC 2 þC ÞE 0:4004 0:8427 6 0:7126 0:5615 ¼6 4 0:3532 0:4957 0:9666 0:9357 24 0 C25 ¼ ðC 2 þC ÞE 0:4004 0:8427 6 0:7126 0:5615 ¼6 4 0:3532 0:4957 0:9667 0:9357 3 0:0000 0:3750 7 7 0:0000 5 0:0000 0:2541 0:2355 0:0906 0:4585 3 0:3160 0:5856 7 7 0:1859 5 0:3509 0:2541 0:2355 0:0906 0:4585 3 0:3160 0:5856 7 7 0:1859 5 0:3509 123 1148 G.-H. Tzeng et al. Appendix 2 Result of the customer evaluation in DEMATAL for the operation processes of Example 2 3 2 1:00 0:00 0:00 0:00 0:00 6 0:00 1:00 0:00 0:00 0:00 7 7 6 7 I¼6 6 0:00 0:00 1:00 0:00 0:00 7 4 0:00 0:00 0:00 1:00 0:00 5 0:00 0:00 0:00 0:00 1:00 3 2 0:0000 0:2222 0:4444 0:1111 0:2222 6 0:0000 0:0000 0:1111 0:0000 0:0000 7 7 6 7 D¼6 6 0:2222 0:3333 0:0000 0:0000 0:4444 7 4 0:0000 0:0000 0:0000 0:0000 0:2222 5 0:0000 0:0000 0:2222 0:3333 0:0000 3 2 1:0000 0:2222 0:4444 0:1111 0:2222 7 6 6 0:0000 1:0000 0:1111 0:0000 0:0000 7 7 6 7 ID¼6 6 0:2222 0:3333 1:0000 0:0000 0:4444 7 7 6 4 0:0000 0:0000 0:0000 1:0000 0:2222 5 0:0000 0:0000 0:2222 0:3333 1:0000 1:1589 6 0:0334 6 6 1 ðI  DÞ ¼ 6 6 0:3007 6 4 0:0160 0:4958 0:7150 0:3461 0:6522 2 1:0575 0:1504 0:0307 0:0811 7 7 7 0:5179 1:3533 0:2766 0:7297 7 7 7 0:0276 0:0722 1:0947 0:2789 5 0:0722 0:1243 0:3248 0:4263 1:2551 1 F ¼ DðI 2  DÞ 0:1589 0:4958 6 0:0334 0:0575 6 6 ¼ 6 0:3007 0:5179 6 4 0:0160 0:0276 0:0722 0:1243 0:7150 0:1504 0:3533 0:0722 0:3248 0:3461 0:0307 0:2766 0:0947 0:4263 3 0:6522 0:0811 7 7 7 0:7297 7 7 0:2789 5 0:2551 Result of the customer evaluation in FDM for the operation processes of Example 2 E¼D 2 0 6 0 6 E¼6 6 0:2222 4 0 0 C0 ¼ I 2 1 60 6 C0 ¼ 6 60 40 0 123 0 1 0 0 0 0:2222 0 0:3333 0 0 0 0 1 0 0 0 0 0 1 0 3 0:4444 0:1111 0:1111 0 0 0 0 0 0:2222 0:3333 3 0 07 7 07 7 05 1 3 0:2222 0 7 7 0:4444 7 7 0:2222 5 0 0 C1 ¼ C 2 E 0 6 0 6 ¼6 6 0:2222 4 0 0 0:2222 0 0:3333 0 0 0:4444 0:1111 0 0 0:2222 0:1111 0 0 0 0:3333 1 0 C2 ¼ ðC 2 þC ÞE 0:0987 0:3703 6 0:0247 0:0370 6 ¼6 6 0:2222 0:3827 4 0 0 0:0494 0:0741 3 0:2222 0 7 7 0:4444 7 7 0:2222 5 0 0:5185 0:1111 0:2345 0:0494 0:2222 0:1852 0 0:1728 0:0741 0:3333 3 0:4444 0:0494 7 7 0:4938 7 7 0:2222 5 0:1728 0:6282 0:1372 0:2510 0:0494 0:2908 0:2702 0:0192 0:1893 0:0741 0:3964 3 0:5157 0:0549 7 7 0:6364 7 7 0:2606 5 0:1838 2 0 C3 ¼ ðC 2 þC ÞE 0:1152 0:4169 6 0:0247 0:0425 6 ¼6 6 0:2743 0:4608 4 0:0110 0:0165 0:0494 0:0850 … C 17 ¼ 2 ðC16 þ C 0 Þ E 0:1589 0:4958 6 0:0334 0:0575 6 ¼6 6 0:3007 0:5179 4 0:0160 0:0276 0:0722 0:1243 C 18 ¼ 2 ðC17 þ C 0 Þ E 0:1589 0:4958 6 0:0334 0:0575 6 ¼6 6 0:3007 0:5179 4 0:0160 0:0276 0:0722 0:1243 0:7150 0:1503 0:3533 0:0722 0:3247 0:3461 0:0307 0:2766 0:0947 0:4263 3 0:6521 0:0811 7 7 0:7297 7 7 0:2789 5 0:2551 0:7150 0:1503 0:3533 0:0722 0:3247 0:3461 0:0307 0:2766 0:0947 0:4263 3 0:6521 0:0811 7 7 0:7297 7 7 0:2789 5 0:2551 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: LS 0 6 0:1923 TPE 6 E¼ 6 RPE 4 0:1538 TW 0:2692 LS 2 TPE 0:3462 0 0:3462 0:3846 RPE 0:1923 0:1538 0 0:1923 TW 3 0:4615 0:1154 7 7 7 0:1538 5 0 Fuzzy decision maps 1149 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 FDM (f(x) = x) LS TPE RPE Leadership (LS) 0.5843a 1.1514a 0.6677 0.5355 a 0.9667a TPE 0.4648 0.4714 0.4014 0.4460 0.5128 0.8462 0.3325 0.5392a TW 0.7039 1.0386 0.5904 0.5355 Means the difference between FDM and DEMATEL is 0.0001 In above table, the numerical results show the DEMATEL method and the FDM method using linear function almost the same. This finding supports the FDM is a general 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: MS CS MGR BL FP 3 0:2903 0:1935 0:2581 0:1613 CS 6 0 0:2258 0:3226 0:1290 7 7 6 0:3226 7 6 E ¼ MGR 6 0:1613 0:0645 0 0:0645 0:2903 7 7 6 7 6 0 0:0968 5 BL 4 0:3226 0:2903 0:1290 MS 2 0 FP 0:0968 0:0323 0:3548 0:0323 0 Next, we can obtain the steady-state matrix by calculating Eq. 6 in the FDM method and Eq. 5 in the DEMATEL as follows. CS MGR 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 TW RPE a continued DEMATEL MS 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 References Andreou AS, Mateou NH, Zombanakis GA (2005) Soft computing for crisis management and political decision making: the use of genetically evolved fuzzy cognitive maps. Soft Comput 9(3): 194–210 Axelrod R (1976) Structure of decision, the cognitive maps of political elite. Princeton University Press, London Chen SJ, Hwang CL (1992) Fuzzy multiple attribute decision making: methods and applications. 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