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Fuzzy decision maps: a generalization of the DEMATEL methods

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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.

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Correspondence to Rachung Yu.

Appendices

Appendix 1

1.1 Result of the supplier evaluation in DEMATAL for the operation processes of Example 1

$$ \user2{I} = \left[ {\begin{array}{*{20}c} {1.00} & {0.00} & {0.00} & {0.00} \\ {0.00} & {1.00} & {0.00} & {0.00} \\ {0.00} & {0.00} & {1.00} & {0.00} \\ {0.00} & {0.00} & {0.00} & {1.00} \\ \end{array} } \right] $$
$$ \user2{D} = \left[ {\begin{array}{*{20}c} {0.000} & {0.500} & {0.125} & {0.000} \\ {0.250} & {0.000} & {0.000} & {0.375} \\ {0.125} & {0.250} & {0.000} & {0.000} \\ {0.500} & {0.250} & {0.250} & {0.000} \\ \end{array} } \right] $$
$$ \user2{I} - \user2{D} = \left[ {\begin{array}{*{20}c} {1.000} & { - 0.500} & { - 0.125} & {0.000} \\ { - 0.250} & {1.000} & {0.000} & { - 0.375} \\ { - 0.125} & { - 0.250} & {1.000} & {0.000} \\ { - 0.500} & { - 0.250} & { - 0.250} & {1.000} \\ \end{array} } \right] $$
$$ (\user2{I} - \user2{D})^{ - 1} = \left[ {\begin{array}{*{20}c} {1.4005} & {0.8428} & {0.2541} & {0.3160} \\ {0.7126} & {1.5616} & {0.2355} & {0.5856} \\ {0.3532} & {0.4957} & {1.0906} & {0.1859} \\ {0.9667} & {0.9357} & {0.4586} & {1.3509} \\ \end{array} } \right] $$
$$ \user2{F} = \user2{D}(\user2{I} - \user2{D})^{ - 1} = \left[ {\begin{array}{*{20}c} {0.4005} & {0.8428} & {0.2541} & {0.3160} \\ {0.7126} & {0.5616} & {0.2355} & {0.5856} \\ {0.3532} & {0.4957} & {0.0906} & {0.1859} \\ {0.9667} & {0.9357} & {0.4586} & {0.3509} \\ \end{array} } \right] $$

1.2 Result of the supplier evaluation in FDM for the operation processes of Example 1

$$ \user2{E} = \user2{D} $$
$$ \user2{E} = \left[ {\begin{array}{*{20}c} {0.0000} & {0.5000} & {0.1250} & {0.0000} \\ {0.2500} & {0.0000} & {0.0000} & {0.3750} \\ {0.1250} & {0.2500} & {0.0000} & {0.0000} \\ {0.5000} & {0.2500} & {0.2500} & {0.0000} \\ \end{array} } \right] $$
$$ \user2{C}^{0} = \user2{I} $$
$$ \user2{C}^{0} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} } \right] $$
$$ \user2{C}^{1} = \user2{C}^{0} \;\user2{E} = \left[ {\begin{array}{*{20}c} {0.0000} & {0.5000} & {0.1250} & {0.0000} \\ {0.2500} & {0.0000} & {0.0000} & {0.3750} \\ {0.1250} & {0.2500} & {0.0000} & {0.0000} \\ {0.5000} & {0.2500} & {0.2500} & {0.0000} \\ \end{array} } \right] $$
$$ \user2{C}^{2} = (\user2{C}^{1} + \user2{C}^{0} )\;\user2{E} = \left[ {\begin{array}{*{20}c} {0.1406} & {0.5313} & {0.1250} & {0.1875} \\ {0.4375} & {0.2188} & {0.1250} & {0.3750} \\ {0.1875} & {0.3125} & {0.0156} & {0.0938} \\ {0.5938} & {0.5625} & {0.3125} & {0.0938} \\ \end{array} } \right] $$
$$ \user2{C}^{3} = (\user2{C}^{2} + \user2{C}^{0} )\;\user2{E} = \left[ {\begin{array}{*{20}c} {0.2422} & {0.6484} & {0.1895} & {0.1992} \\ {0.5078} & {0.3438} & {0.1484} & {0.4570} \\ {0.2520} & {0.3711} & {0.0469} & {0.1172} \\ {0.7266} & {0.6484} & {0.3477} & {0.2109} \\ \end{array} } \right] $$

$$ \user2{C}^{24} = (\user2{C}^{23} + \user2{C}^{0} )\;\user2{E} = \left[ {\begin{array}{*{20}c} {0.4004} & {0.8427} & {0.2541} & {0.3160} \\ {0.7126} & {0.5615} & {0.2355} & {0.5856} \\ {0.3532} & {0.4957} & {0.0906} & {0.1859} \\ {0.9666} & {0.9357} & {0.4585} & {0.3509} \\ \end{array} } \right] $$
$$ \user2{C}^{25} = (\user2{C}^{24} + \user2{C}^{0} )\;\user2{E} = \left[ {\begin{array}{*{20}c} {0.4004} & {0.8427} & {0.2541} & {0.3160} \\ {0.7126} & {0.5615} & {0.2355} & {0.5856} \\ {0.3532} & {0.4957} & {0.0906} & {0.1859} \\ {0.9667} & {0.9357} & {0.4585} & {0.3509} \\ \end{array} } \right] $$

Appendix 2

2.1 Result of the customer evaluation in DEMATAL for the operation processes of Example 2

$$ \user2{I} = \left[ {\begin{array}{*{20}c} {1.00} & {0.00} & {0.00} & {0.00} & {0.00} \\ {0.00} & {1.00} & {0.00} & {0.00} & {0.00} \\ {0.00} & {0.00} & {1.00} & {0.00} & {0.00} \\ {0.00} & {0.00} & {0.00} & {1.00} & {0.00} \\ {0.00} & {0.00} & {0.00} & {0.00} & {1.00} \\ \end{array} } \right] $$
$$ \user2{D} = \left[ {\begin{array}{*{20}c} {0.0000} & {0.2222} & {0.4444} & {0.1111} & {0.2222} \\ {0.0000} & {0.0000} & {0.1111} & {0.0000} & {0.0000} \\ {0.2222} & {0.3333} & {0.0000} & {0.0000} & {0.4444} \\ {0.0000} & {0.0000} & {0.0000} & {0.0000} & {0.2222} \\ {0.0000} & {0.0000} & {0.2222} & {0.3333} & {0.0000} \\ \end{array} } \right] $$
$$ \user2{I} - \user2{D} = \left[ {\begin{array}{*{20}c} {1.0000} & { - 0.2222} & { - 0.4444} & { - 0.1111} & { - 0.2222} \\ {0.0000} & {1.0000} & { - 0.1111} & {0.0000} & {0.0000} \\ { - 0.2222} & { - 0.3333} & {1.0000} & {0.0000} & { - 0.4444} \\ {0.0000} & {0.0000} & {0.0000} & {1.0000} & { - 0.2222} \\ {0.0000} & {0.0000} & { - 0.2222} & { - 0.3333} & {1.0000} \\ \end{array} } \right] $$
$$ (\user2{I} - \user2{D})^{ - 1} = \left[ {\begin{array}{*{20}c} {1.1589} & {0.4958} & {0.7150} & {0.3461} & {0.6522} \\ {0.0334} & {1.0575} & {0.1504} & {0.0307} & {0.0811} \\ {0.3007} & {0.5179} & {1.3533} & {0.2766} & {0.7297} \\ {0.0160} & {0.0276} & {0.0722} & {1.0947} & {0.2789} \\ {0.0722} & {0.1243} & {0.3248} & {0.4263} & {1.2551} \\ \end{array} } \right] $$
$$ \user2{F} = \user2{D}(\user2{I} - \user2{D})^{ - 1} = \left[ {\begin{array}{*{20}c} {0.1589} & {0.4958} & {0.7150} & {0.3461} & {0.6522} \\ {0.0334} & {0.0575} & {0.1504} & {0.0307} & {0.0811} \\ {0.3007} & {0.5179} & {0.3533} & {0.2766} & {0.7297} \\ {0.0160} & {0.0276} & {0.0722} & {0.0947} & {0.2789} \\ {0.0722} & {0.1243} & {0.3248} & {0.4263} & {0.2551} \\ \end{array} } \right] $$

2.2 Result of the customer evaluation in FDM for the operation processes of Example 2

$$ \user2{E} = \user2{D} $$
$$ \user2{E} = \left[ {\begin{array}{*{20}c} 0 & {0.2222} & {0.4444} & {0.1111} & {0.2222} \\ 0 & 0 & {0.1111} & 0 & 0 \\ {0.2222} & {0.3333} & 0 & 0 & {0.4444} \\ 0 & 0 & 0 & 0 & {0.2222} \\ 0 & 0 & {0.2222} & {0.3333} & 0 \\ \end{array} } \right] $$
$$ \user2{C}^{\user2{0}} = \user2{I} $$
$$ \user2{C}^{0} = \left[ {\begin{array}{*{20}c} 1 & 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 1 \\ \end{array} } \right] $$
$$ \user2{C}^{1} = \user2{C}^{0} \times \user2{E} = \left[ {\begin{array}{*{20}c} 0 & {0.2222} & {0.4444} & {0.1111} & {0.2222} \\ 0 & 0 & {0.1111} & 0 & 0 \\ {0.2222} & {0.3333} & 0 & 0 & {0.4444} \\ 0 & 0 & 0 & 0 & {0.2222} \\ 0 & 0 & {0.2222} & {0.3333} & 0 \\ \end{array} } \right] $$
$$ \user2{C}^{2} = (\user2{C}^{1} + \user2{C}^{0} )\;\user2{E} = \left[ {\begin{array}{*{20}c} {0.0987} & {0.3703} & {0.5185} & {0.1852} & {0.4444} \\ {0.0247} & {0.0370} & {0.1111} & 0 & {0.0494} \\ {0.2222} & {0.3827} & {0.2345} & {0.1728} & {0.4938} \\ 0 & 0 & {0.0494} & {0.0741} & {0.2222} \\ {0.0494} & {0.0741} & {0.2222} & {0.3333} & {0.1728} \\ \end{array} } \right] $$
$$ \user2{C}^{3} = (\user2{C}^{2} + \user2{C}^{0} )\;\user2{E} = \left[ {\begin{array}{*{20}c} {0.1152} & {0.4169} & {0.6282} & {0.2702} & {0.5157} \\ {0.0247} & {0.0425} & {0.1372} & {0.0192} & {0.0549} \\ {0.2743} & {0.4608} & {0.2510} & {0.1893} & {0.6364} \\ {0.0110} & {0.0165} & {0.0494} & {0.0741} & {0.2606} \\ {0.0494} & {0.0850} & {0.2908} & {0.3964} & {0.1838} \\ \end{array} } \right] $$

$$ C^{ 1 7} = (C^{ 1 6} + C^{ 0} )\;E = \left[ {\begin{array}{*{20}c} {0.1589} & {0.4958} & {0.7150} & {0.3461} & {0.6521} \\ {0.0334} & {0.0575} & {0.1503} & {0.0307} & {0.0811} \\ {0.3007} & {0.5179} & {0.3533} & {0.2766} & {0.7297} \\ {0.0160} & {0.0276} & {0.0722} & {0.0947} & {0.2789} \\ {0.0722} & {0.1243} & {0.3247} & {0.4263} & {0.2551} \\ \end{array} } \right] $$
$$ C^{ 1 8} = (C^{ 1 7} + C^{ 0} )\;E = \left[ {\begin{array}{*{20}c} {0.1589} & {0.4958} & {0.7150} & {0.3461} & {0.6521} \\ {0.0334} & {0.0575} & {0.1503} & {0.0307} & {0.0811} \\ {0.3007} & {0.5179} & {0.3533} & {0.2766} & {0.7297} \\ {0.0160} & {0.0276} & {0.0722} & {0.0947} & {0.2789} \\ {0.0722} & {0.1243} & {0.3247} & {0.4263} & {0.2551} \\ \end{array} } \right] $$

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:

$$ \user2{E} = \begin{array}{*{20}c} {} & {{\text{LS}}\quad \quad {\text{TPE}}\quad \quad {\text{RPE}}\quad \quad {\text{TW}}} \\ \begin{gathered} {\text{LS}} \hfill \\ {\text{TPE}} \hfill \\ {\text{RPE}} \hfill \\ {\text{TW}} \hfill \\ \end{gathered} & {\left[ {\begin{array}{*{20}c} 0 & {0.3462} & {0.1923} & {0.4615} \\ {0.1923} & 0 & {0.1538} & {0.1154} \\ {0.1538} & {0.3462} & 0 & {0.1538} \\ {0.2692} & {0.3846} & {0.1923} & 0 \\ \end{array} } \right]} \\ \end{array} $$

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

  1. aMeans 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:

$$ \user2{E} = \begin{array}{*{20}c} {} & {{\text{MS}}\quad \quad {\text{CS}}\quad \quad {\text{MGR}}\quad \quad {\text{BL}}\quad \quad {\text{FP}}} \\ \begin{gathered} {\text{MS}} \hfill \\ {\text{CS}} \hfill \\ {\text{MGR}} \hfill \\ {\text{BL}} \hfill \\ {\text{FP}} \hfill \\ \end{gathered} & {\left[ {\begin{array}{*{20}c} 0 & {0.2903} & {0.1935} & {0.2581} & {0.1613} \\ {0.3226} & 0 & {0.2258} & {0.3226} & {0.1290} \\ {0.1613} & {0.0645} & 0 & {0.0645} & {0.2903} \\ {0.3226} & {0.2903} & {0.1290} & 0 & {0.0968} \\ {0.0968} & {0.0323} & {0.3548} & {0.0323} & 0 \\ \end{array} } \right]} \\ \end{array} $$

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

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

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

  1. 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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Tzeng, GH., Chen, WH., Yu, R. et al. Fuzzy decision maps: a generalization of the DEMATEL methods. Soft Comput 14, 1141–1150 (2010). https://doi.org/10.1007/s00500-009-0507-0

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