A new DEMATEL method based on interval type-2 fuzzy sets for developing causal relationship of knowledge management criteria

Abstract

The fuzzy decision-making trial and evaluation laboratory (fuzzy DEMATEL) has been used to solve various multi-criteria group decision-making problems where triangular type-1 fuzzy sets are utilized in defining decision makers’ linguistic evaluation. Most of the fuzzy DEMATEL modifications are built from linguistic variables based on type-1 fuzzy sets (T1FS). Previous literature suggests that interval type-2 fuzzy sets (IT2FS) can offer an alternative that can handle vagueness and uncertainty. This paper proposes a modification fuzzy DEMATEL characterized by IT2FS for linguistic variables. Differently from the typical fuzzy DEMATEL which directly utilizes triangular type-1 fuzzy numbers, this modification introduces trapezoidal IT2 fuzzy numbers to enhance evaluation in the group decision-making environment. This new modification includes linguistic variables expressed by IT2FS and an expected value method for normalizing upper and lower memberships of IT2FS to crisp numbers. The proposed modification is applied to a case of knowledge management (KM) where eleven criteria are considered. Three experts in KM were invited to provide linguistic judgments with respect to the criteria, and the eight-step computational procedure of the proposed modification was implemented without losing the originality of the DEMATEL method. The results unveiled that ‘trust’ is the most influential criteria in KM. Therefore, trust is a phenomenon that impacts on the success of KM. Comparable results are also presented to check the feasibility of the proposed method. It is shown that the criteria weight and the causal relationship of criteria using the proposed method are consistent with the other two methods.

Appendices

Appendix 1

1.1 Trapezoidal fuzzy numbers

A trapezoidal fuzzy number can be defined as $\tilde{m} = \left( {a,b,c,d} \right)$ where the membership functions $\mu_{{\tilde{m}}}$ of $\tilde{m}$ is given by:

$$ \mu_{{\tilde{m}}} = \left\{ {\begin{array}{*{20}c} {\frac{x - a}{b - a}} & {\left( {a \le x \le b} \right)} \\ 1 & {\left( {b \le x \le c} \right)} \\ {\frac{d - x}{d - c}} & {\left( {c \le x \le d} \right)} \\ \end{array} } \right. $$

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where b and c are called a mode interval of $\tilde{m}$, a and d are called lower and upper limits of $\tilde{m}$, respectively [35].

Let $\tilde{A}$ and $\tilde{B}$ be two positive trapezoidal fuzzy numbers parameterized by $\left( {a_{1} ,a_{2} ,a_{3} ,a_{4} } \right)$ and $\left( {b_{1} ,b_{2} ,b_{3} ,b_{4} } \right)$, then the arithmetic operations of these two trapezoidal fuzzy numbers are given as follows [6].

$$ \tilde{A} \oplus \tilde{B} = \left( {a_{1} + b_{1} ,a_{2} + b_{2} ,a_{3} + b_{3} ,a_{4} + b_{4} } \right) $$

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$$ \tilde{A} - \tilde{B} = \left( {a_{1} - b_{1} ,a_{2} - b_{2} ,a_{3} - b_{3} ,a_{4} - b_{4} } \right) $$

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$$ \tilde{A} \otimes \tilde{B} = \left( {a_{1} \times b_{1} ,a_{2} \times b_{2} ,a_{3} \otimes b_{3} ,a_{4} \times b_{4} } \right) $$

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$$ \left( {\tilde{A}} \right)^{ - 1} = \left( {\frac{1}{{a_{4} }},\frac{1}{{a_{3} }},\frac{1}{{a_{2} }},\frac{1}{{a_{1} }}} \right) $$

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1.2 Type-1 fuzzy set

Let $\tilde{A}$ be a type-1 trapezoidal fuzzy set, $\tilde{A} = \left( {a_{1} ,a_{2} ,a_{3} ,a_{4} ;H_{1} \left( A \right),H_{2} \left( A \right)} \right)$. Figure 2 shows the $\tilde{A}$ where $H_{1} \left( {\tilde{A}} \right)$ denotes the membership value of the element $a_{2} ,H_{2} \left( {\tilde{A}} \right)$ denotes the membership value of the element $a_{3} ,0 \le H_{1} \left( A \right) \le 1$ and $0 \le H_{2} \left( A \right) \le 1$. If $a_{2} = a_{3}$, then the type-1 fuzzy set $\tilde{A}$ becomes a triangular T1FS.

Fig. 2

A type-1 trapezoidal fuzzy set

1.3 Interval type-2 fuzzy set

We briefly present some definitions of T2FS and IT2 FS. Mendel et al. [30] proposed the following definitions of T2FS.

Definition 1.1

A type-2 fuzzy set $\tilde{\tilde{A}}$ in the universe of discourse X can be represented by a type-2 membership function ${\mu_{{\tilde{\tilde{A}}}}}$ shown as follows:

$$ \tilde{\tilde{A}} = \left\{ {\left( {x,u} \right),\mu_{{\tilde{\tilde{A}}}} \left( {x,u} \right)\left| {\forall x \in X,\forall u \in J_{x} \subseteq \left[ {0,1} \right],0 \le \mu_{{\tilde{\tilde{A}}}} \left( {x,u} \right) \le 1} \right.} \right\} $$

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where J_x denotes an interval in [0, 1].

The type-2 fuzzy set $\tilde{\tilde{A}}$ also can be written as follows.

Definition 1.2

A type-2 fuzzy set $\tilde{\tilde{A}}$ in the universe of discourse X can be represented by a type-2 membership function $\mu_{{\tilde{\tilde{A}}}}$.

$$ \tilde{\tilde{A}} = \int_{x \in X} {\int_{{u \in J_{x} }} {\mu_{{\tilde{\tilde{A}}}} } } \left( {x,u} \right)/\left( {x,u} \right) $$

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where $J_{x} \subseteq \left[ {0,1} \right]$ and $\int {\int {} }$ denotes the union over all admissible x and u.

For simplicity, the T2FS $\tilde{\tilde{A}}$ may be written as interval membership.

Definition 1.3

Let $\tilde{\tilde{A}}$ be a T2FS in the universe of discourse X represented by the type-2 membership function $\mu_{{\tilde{\tilde{A}}}}$. If all $\mu_{{\tilde{\tilde{A}}}} \left( {x,u} \right) = 1$, then $\tilde{\tilde{A}}$ is called IT2 FS. An IT2 FS $\tilde{\tilde{A}}$ can be regarded as a special case of T2FS, shown as follows:

$$ \tilde{\tilde{A}} = \int_{x \in X} {\int_{{u \in J_{x} }} {1/\left( {x,u} \right),} } $$

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where $J_{x} \subseteq \left[ {0,1} \right]$.

The property of interval in defining the T2FS paves a way to introduce boundary membership of upper and lower. The upper and lower memberships are defined as follows.

Definition 1.4

The upper membership function (UMF) and lower membership function (LMF) of $\mathop {\text{A}}\limits^{ \approx }$ are two type-1 membership functions.

The heights of the UMF and LMF of IT2 FS are also defined to characterize IT2 FS. Figure 3 shows trapezoidal IT2 FS where upper and lower fuzzy numbers are drawn as reference points.

Fig. 3

An interval type-2 trapezoidal fuzzy set

Figure 2 shows the upper trapezoidal membership function $\tilde{A}_{i}^{U}$ and the lower trapezoidal membership function $\tilde{A}_{i}^{L}$ of IT2 FS $\tilde{A}_{i}$.

1.4 Arithmetic operations of trapezoidal interval type-2 fuzzy sets

Arithmetic operations of trapezoidal IT2FSs are described by [42]. It is recalled as follows.

Definition 1.5

The addition operation between the trapezoidal IT2FS

$$ \tilde{\tilde{A}}_{1} = \left( {\tilde{A}_{1}^{U} ,\tilde{A}_{1`}^{L} } \right) = \left( {\left( {a_{11}^{U} ,a_{12}^{U} ,a_{13}^{U} ,a_{14}^{U} ;H_{1} \left( {\tilde{A}_{1}^{U} } \right),H_{2} \left( {\tilde{A}_{1}^{U} } \right)} \right),\left( {a_{11}^{L} ,a_{12}^{L} ,a_{13}^{L} ,a_{14}^{L} ;H_{1} \left( {\tilde{A}_{1}^{L} } \right),H_{2} \left( {\tilde{A}_{1}^{L} } \right)} \right)} \right) $$

and

$$ \tilde{\tilde{A}}_{2} = \left( {\tilde{A}_{2}^{U} ,\tilde{A}_{2}^{L} } \right) = \left( {\left( {a_{21}^{U} ,a_{22}^{U} ,a_{23}^{U} ,a_{24}^{U} ;H_{1} \left( {\tilde{A}_{2}^{U} } \right),H_{2} \left( {\tilde{A}_{2}^{U} } \right)} \right),\left( {a_{21}^{L} ,a_{22}^{L} ,a_{23}^{L} ,a_{24}^{L} ;H_{1} \left( {\tilde{A}_{2}^{L} } \right),H_{2} \left( {\tilde{A}_{2}^{L} } \right)} \right)} \right) $$

is defined as follows:

$$ \begin{aligned} &\tilde{\tilde{A}}_{1} \oplus \tilde{\tilde{A}}_{2} \hfill \\ &\quad= \left( {\tilde{A}_{1}^{U} ,\tilde{A}_{1}^{L} } \right) \oplus \left( {\tilde{A}_{2}^{U} ,\tilde{A}_{2}^{L} } \right) \hfill \\ &\quad= \left( {\left( {a_{11}^{U} + a_{21}^{U} ,a_{12}^{U} + a_{22}^{U} ,a_{13}^{U} + a_{23}^{U} ,a_{14}^{U} + a_{24}^{U} ;\hbox{min} \left( {H_{1} \left( {\tilde{A}_{1}^{U} } \right),H_{2} \left( {\tilde{A}_{2}^{U} } \right)} \right),\hbox{min} \left( {H_{2} \left( {\tilde{A}_{1}^{U} } \right),H_{2} \left( {\tilde{A}_{2}^{U} } \right)} \right)} \right)} \right), \hfill \\ &\quad\left( {a_{11}^{L} + a_{21}^{L} ,a_{12}^{L} + a_{22}^{L} ,a_{13}^{L} + a_{23}^{L} ,a_{14}^{L} + a_{24}^{L} ;\hbox{min} \left( {H_{1} \left( {\tilde{A}_{1}^{L} } \right),H_{2} \left( {\tilde{A}_{2}^{L} } \right)} \right),\hbox{min} \left( {H_{2} \left( {\tilde{A}_{1}^{L} } \right),H_{2} \left( {\tilde{A}_{2}^{L} } \right)} \right)} \right) \hfill \\ \end{aligned} $$

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Definition 1.6

The subtraction operation between the trapezoidal IT2FS

$$ \tilde{\tilde{A}}_{1} = \left( {\tilde{A}_{1}^{U} ,\tilde{A}_{1`}^{L} } \right) = \left( {\left( {a_{11}^{U} ,a_{12}^{U} ,a_{13}^{U} ,a_{14}^{U} ;H_{1} \left( {\tilde{A}_{1}^{U} } \right),H_{2} \left( {\tilde{A}_{1}^{U} } \right)} \right),\left( {a_{11}^{L} ,a_{12}^{L} ,a_{13}^{L} ,a_{14}^{L} ;H_{1} \left( {\tilde{A}_{1}^{L} } \right),H_{2} \left( {\tilde{A}_{1}^{L} } \right)} \right)} \right) $$

and

$$ \tilde{\tilde{A}}_{2} = \left( {\tilde{A}_{2}^{U} ,\tilde{A}_{2}^{L} } \right) = \left( {\left( {a_{21}^{U} ,a_{22}^{U} ,a_{23}^{U} ,a_{24}^{U} ;H_{1} \left( {\tilde{A}_{2}^{U} } \right),H_{2} \left( {\tilde{A}_{2}^{U} } \right)} \right),\left( {a_{21}^{L} ,a_{22}^{L} ,a_{23}^{L} ,a_{24}^{L} ;H_{1} \left( {\tilde{A}_{2}^{L} } \right),H_{2} \left( {\tilde{A}_{2}^{L} } \right)} \right)} \right) $$

is defined as follows:

$$ \begin{aligned} &\tilde{\tilde{A}}_{1} - \tilde{\tilde{A}}_{2} \hfill \\ &\quad= \left( {\tilde{A}_{1}^{U} ,\tilde{A}_{1}^{L} } \right) - \left( {\tilde{A}_{2}^{U} ,\tilde{A}_{2}^{L} } \right) \hfill \\ &\quad= \left( {\left( {a_{11}^{U} - a_{24}^{U} ,a_{12}^{U} - a_{23}^{U} ,a_{13}^{U} - a_{22}^{U} ,a_{14}^{U} - a_{21}^{U} ;\hbox{min} \left( {H_{1} \left( {\tilde{A}_{1}^{U} } \right),H_{1} \left( {\tilde{A}_{2}^{U} } \right),\hbox{min} \left( {H_{2} \left( {\tilde{A}_{1}^{U} } \right),H_{2} \left( {\tilde{A}_{2}^{U} } \right)} \right.} \right)} \right)} \right., \hfill \\ &\quad\left( {a_{11}^{L} - a_{24}^{L} ,a_{12}^{L} - a_{23}^{L} ,a_{13}^{L} - a_{22}^{L} ,a_{14}^{L} - a_{21}^{L} ;\hbox{min} \left( {H_{1} \left( {\tilde{A}_{1}^{L} } \right),H_{1} \left( {\tilde{A}_{2}^{L} } \right),\hbox{min} \left( {H_{2} \left( {\tilde{A}_{1}^{L} } \right),H_{2} \left( {\tilde{A}_{2}^{L} } \right)} \right.} \right)} \right) \hfill \\ \end{aligned} $$

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Definition 1.7

The multiplication operation between the trapezoidal IT2FS

$$ \tilde{\tilde{A}}_{1} = \left( {\tilde{A}_{1}^{U} ,\tilde{A}_{1`}^{L} } \right) = \left( {\left( {a_{11}^{U} ,a_{12}^{U} ,a_{13}^{U} ,a_{14}^{U} ;H_{1} \left( {\tilde{A}_{1}^{U} } \right),H_{2} \left( {\tilde{A}_{1}^{U} } \right)} \right),\left( {a_{11}^{L} ,a_{12}^{L} ,a_{13}^{L} ,a_{14}^{L} ;H_{1} \left( {\tilde{A}_{1}^{L} } \right),H_{2} \left( {\tilde{A}_{1}^{L} } \right)} \right)} \right) $$

and

$$ \tilde{\tilde{A}}_{2} = \left( {\tilde{A}_{2}^{U} ,\tilde{A}_{2}^{L} } \right) = \left( {\left( {a_{21}^{U} ,a_{22}^{U} ,a_{23}^{U} ,a_{24}^{U} ;H_{1} \left( {\tilde{A}_{2}^{U} } \right),H_{2} \left( {\tilde{A}_{2}^{U} } \right)} \right),\left( {a_{21}^{L} ,a_{22}^{L} ,a_{23}^{L} ,a_{24}^{L} ;H_{1} \left( {\tilde{A}_{2}^{L} } \right),H_{2} \left( {\tilde{A}_{2}^{L} } \right)} \right)} \right) $$

is defined as follows:

$$ \begin{aligned} &\tilde{\tilde{A}}_{1} \otimes \tilde{\tilde{A}}_{2} \hfill \\ &\quad= \left( {\tilde{A}_{1}^{U} ,\tilde{A}_{1}^{L} } \right) \otimes \left( {\tilde{A}_{2}^{U} ,\tilde{A}_{2}^{L} } \right) \hfill \\ &\quad= \left( {\left( {a_{11}^{U} \times a_{21}^{U} ,a_{12}^{U} \times a_{22}^{U} ,a_{13}^{U} \times a_{23}^{U} ,a_{14}^{U} \times a_{24}^{U} ;\hbox{min} \left( {H_{1} \left( {\tilde{A}_{1}^{U} } \right),H_{1} \left( {\tilde{A}_{2}^{U} } \right),\hbox{min} \left( {H_{2} \left( {\tilde{A}_{1}^{U} } \right),H_{2} \left( {\tilde{A}_{2}^{U} } \right)} \right.} \right)} \right)} \right., \hfill \\ &\quad\left( {a_{11}^{L} \times a_{21}^{L} ,a_{12}^{L} \times a_{22}^{L} ,a_{13}^{L} \times a_{23}^{L} ,a_{14}^{L} \times a_{24}^{L} ;\hbox{min} \left( {H_{1} \left( {\tilde{A}_{1}^{L} } \right),H_{1} \left( {\tilde{A}_{2}^{L} } \right),\hbox{min} \left( {H_{2} \left( {\tilde{A}_{1}^{L} } \right),H_{2} \left( {\tilde{A}_{2}^{L} } \right)} \right.} \right)} \right) \hfill \\ \end{aligned} $$

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The above definitions and arithmetic operations are prevalently employed in the proposed IT2 fuzzy DEMATEL.

Appendix 2

2.1 Appendix 2.1

Evaluation of C1 by DM1

DM1	C1
C1	0
C2	((0.8,0.9,0.9,1.0;1,1), (0.85,0.9,0.9,0.95;0.9,0.9))
C3	((0.4,0.5,0.5,0.6;1,1), (0.45,0.5,0.5,0.55;0.9,0.9))
C4	((0.4,0.5,0.5,0.6;1,1), (0.45,0.5,0.5,0.55;0.9,0.9))
C5	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C6	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C7	((0.8,0.9,0.9,1.0;1,1), (0.85,0.9,0.9,0.95;0.9,0.9))
C8	((0.8,0.9,0.9,1.0;1,1), (0.85,0.9,0.9,0.95;0.9,0.9))
C9	((0.8,0.9,0.9,1.0;1,1), (0.85,0.9,0.9,0.95;0.9,0.9))
C10	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C11	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))

2.2 Appendix 2.2

Evaluation of C1 by DM2

DM2	C1
C1	0
C2	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C3	((0.2,0.3,0.3,0.4;1,1), (0.25,0.3,0.3,0.35;0.9,0.9))
C4	((0.4,0.5,0.5,0.6;1,1), (0.45,0.5,0.5,0.55;0.9,0.9))
C5	((0.8,0.9,0.9,1.0;1,1), (0.85,0.9,0.9,0.95;0.9,0.9))
C6	((0.4,0.5,0.5,0.6;1,1), (0.45,0.5,0.5,0.55;0.9,0.9))
C7	((0.8,0.9,0.9,1.0;1,1), (0.85,0.9,0.9,0.95;0.9,0.9))
C8	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C9	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C10	((0.2,0.3,0.3,0.4;1,1), (0.25,0.3,0.3,0.35;0.9,0.9))
C11	((0.4,0.5,0.5,0.6;1,1), (0.45,0.5,0.5,0.55;0.9,0.9))

2.3 Appendix 2.3

Evaluation of C1 by DM3

DM3	C1
C1	0
C2	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C3	((0.4,0.5,0.5,0.6;1,1), (0.45,0.5,0.5,0.55;0.9,0.9))
C4	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C5	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C6	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C7	((0.8,0.9,0.9,1.0;1,1), (0.85,0.9,0.9,0.95;0.9,0.9))
C8	((0.6,0.7,0.7,0.8;1,1), (0.65,0.7,0.7,0.75;0.9,0.9))
C9	((0.8,0.9,0.9,1.0;1,1), (0.85,0.9,0.9,0.95;0.9,0.9))
C10	((0.4,0.5,0.5,0.6;1,1), (0.45,0.5,0.5,0.55;0.9,0.9))
C11	((0.4,0.5,0.5,0.6;1,1), (0.45,0.5,0.5,0.55;0.9,0.9))

Appendix 3

Initial direct-relation matrix, A

	C1								C2								C3
C1	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))
C2	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.6	0.7	0.7	0.8)	(0.65	0.7	0.7	0.75))
C3	((0.33	0.43	0.43	0.53)	(0.38	0.433	0.433	0.48))	((0.6	0.7	0.7	0.8)	(0.65	0.7	0.7	0.75))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))
C4	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.47	0.57	0.57	0.67)	(0.517	0.57	0.57	0.62))
C5	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.4	0.5	0.5	0.6)	(0.45	0.5	0.5	0.55))
C6	((0.53	0.63	0.63	0.53)	(0.58	0.63	0.63	0.68))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.4	0.5	0.5	0.6)	(0.45	0.5	0.5	0.55))
C7	((0.8	0.9	0.9	1)	(0.85	0.9	0.9	0.95))	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.67	0.77	0.77	0.87)	(0.716	0.77	0.77	0.82))
C8	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.33	0.43	0.43	0.53)	(0.38	0.433	0.433	0.48))	((0.33	0.43	0.43	0.53)	(0.383	0.433	0.433	0.48)0
C9	((0.73	0.83	0.83	0.93)	(0.78	0.83	0.83	0.88))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.4	0.5	0.5	0.6)	(0.45	0.5	0.5	0.55))
C10	((0.4	0.5	0.5	0.6)	(0.45	0.5	0.5	0.55))	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.47	0.57	0.57	0.67)	(0.517	0.57	0.57	0.62))
C11	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.53	0.63	0.63	0.53)	(0.583	0.63	0.63	0.68))

	C4								C5								C6
C1	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.60	0.70	0.70	0.80)	((0.65	0.70	0.70	0.75))
C2	((0.80	0.90	0.90	1.00)	(0.85	0.90	0.90	0.95))	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.40	0.50	0.50	0.60)	((0.45	0.50	0.50	0.55))
C3	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))	((0.40	0.50	0.50	0.60)	((0.45	0.50	0.50	0.55))
C4	((0`00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))	((0.40	0.50	0.50	0.60)	((0.45	0.50	0.50	0.55))
C5	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))	((0`00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.47	0.57	0.57	0.67)	((0.52	0.57	0.57	0.62))
C6	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))	((0`00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))
C7	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.47	0.57	0.57	0.67)	((0.52	0.57	0.57	0.62))
C8	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))	((0.73	0.83	0.83	0.93)	(0.78	0.83	0.83	0.88))	((0.60	0.70	0.70	0.80)	((0.65	0.70	0.70	0.75))
C9	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.40	0.50	0.50	0.60)	((0.45	0.50	0.50	0.55))
C10	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.27	0.37	0.37	0.47)	(0.32	0.37	0.37	0.42))	((0.60	0.70	0.70	0.80)	((0.65	0.70	0.70	0.75))
C11	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))	((0.33	0.43	0.43	0.53)	((0.38	0.43	0.43	0.48))

	C7								C8								C9
C1	((0.73	0.83	0.83	0.93)	(0.78	0.83	0.83	0.88))	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))
C2	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))
C3	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))	((0.27	0.37	0.37	0.47)	(0.32	0.37	0.37	0.42))	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))
C4	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))
C5	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))
C6	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))
C7	((0`00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))
C8	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0`00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))
C9	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0`00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))
C10	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))
C11	((0.80	0.90	0.90	1.00)	(0.85	0.90	0.90	0.95))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))

	C10								C11
C1	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))
C2	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))
C3	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))
C4	((0.67	0.77	0.77	0.87)	(0.72	0.77	0.77	0.82))	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))
C5	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))	((0.40	0.50	0.50	0.60)	(0.45	0.50	0.50	0.55))
C6	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))
C7	((0.47	0.57	0.57	0.67)	(0.52	0.57	0.57	0.62))	((0.80	0.90	0.90	1.00)	(0.85	0.90	0.90	0.95))
C8	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))
C9	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))	((0.33	0.43	0.43	0.53)	(0.38	0.43	0.43	0.48))
C10	((0`00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.60	0.70	0.70	0.80)	(0.65	0.70	0.70	0.75))
C11	((0.53	0.63	0.63	0.73)	(0.58	0.63	0.63	0.68))	((0`00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))

Appendix 4

Normalized initial direct-relation matrix, D

	C1								C2								C3
C1	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))
C2	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C3	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))
C4	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))
C5	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))
C6	((0.07	0.08	0.08	0.07)	(0.07	0.08	0.08	0.09))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))
C7	((0.10	0.11	0.11	0.13)	(0.11	0.11	0.11	0.12))	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))
C8	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))
C9	((0.09	0.10	0.10	0.12)	(0.10	0.10	0.10	0.11))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))
C10	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))
C11	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.07	0.08	0.08	0.07)	(0.07	0.08	0.08	0.09))

	C4								C5								C6
C1	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C2	((0.10	0.11	0.11	0.13)	(0.11	0.11	0.11	0.12))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))
C3	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))
C4	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))
C5	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))
C6	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))
C7	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))
C8	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))	((0.09	0.10	0.10	0.12)	(0.10	0.10	0.10	0.11))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C9	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))
C10	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.03	0.05	0.05	0.06)	(0.04	0.05	0.05	0.05))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C11	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))

	C7								C8								C9
C1	((0.09	0.10	0.10	0.12)	(0.10	0.10	0.10	0.11))	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))
C2	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C3	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))	((0.03	0.05	0.05	0.06)	(0.04	0.05	0.05	0.05))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))
C4	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))
C5	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))
C6	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C7	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C8	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C9	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))
C10	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))
C11	((0.10	0.11	0.11	0.13)	(0.11	0.11	0.11	0.12))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))

	C10								C11
C1	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))
C2	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))
C3	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))
C4	((0.08	0.10	0.10	0.11)	(0.09	0.10	0.10	0.10))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))
C5	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))	((0.05	0.06	0.06	0.08)	(0.06	0.06	0.06	0.07))
C6	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))
C7	((0.06	0.07	0.07	0.08)	(0.07	0.07	0.07	0.08))	((0.10	0.11	0.11	0.13)	(0.11	0.11	0.11	0.12))
C8	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))
C9	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))	((0.04	0.05	0.05	0.07)	(0.05	0.05	0.05	0.06))
C10	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))	((0.08	0.09	0.09	0.10)	(0.08	0.09	0.09	0.09))
C11	((0.07	0.08	0.08	0.09)	(0.07	0.08	0.08	0.09))	((0.00	0.00	0.00	0.00)	(0.00	0.00	0.00	0.00))