6 Dynamic fuzzy hierarchical relationships

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$(({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e}) \land ({\overset{\leftarrow}{x}}_{i k}, {\vec{x}}_{i k}) \land ({\overset{\leftarrow}{r}}_{k j}, {\vec{r}}_{k j})) \leq (({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e}) \land ({\overset{\leftarrow}{r}}_{k j}, {\vec{r}}_{k j})) \leq ({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j}),$ $(({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e}) \land ({\overset{\leftarrow}{x}}_{i k}, {\vec{x}}_{i k}) \land ({\overset{\leftarrow}{r}}_{k j}, {\vec{r}}_{k j})) \leq (({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e}) \land ({\overset{\leftarrow}{r}}_{k j}, {\vec{r}}_{k j})) \leq ({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j}),$

and when $(({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e}) \land ({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e})) > ({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j})$ $(({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e}) \land ({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e})) > ({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j})$ is true, there is

$(({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e}) \land ({\overset{\leftarrow}{x}}_{e k}, {\vec{x}}_{e k}) \land ({\overset{\leftarrow}{r}}_{k j}, \vec{r_{k j}})) \leq ({\overset{\leftarrow}{x}}_{e l}, {\vec{x}}_{e k}) \leq ({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j}) .$ $(({\overset{\leftarrow}{r}}_{i e}, {\vec{r}}_{i e}) \land ({\overset{\leftarrow}{x}}_{e k}, {\vec{x}}_{e k}) \land ({\overset{\leftarrow}{r}}_{k j}, \vec{r_{k j}})) \leq ({\overset{\leftarrow}{x}}_{e l}, {\vec{x}}_{e k}) \leq ({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j}) .$

Hence, $({\overset{\leftarrow}{u}}_{i j}, {\vec{u}}_{i j}) \leq ({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j}) (i \leq n, j \leq m);$ $({\overset{\leftarrow}{u}}_{i j}, {\vec{u}}_{i j}) \leq ({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j}) (i \leq n, j \leq m);$ that is,

$(\overset{\leftarrow}{R}, \vec{R}) \circ (\overset{\leftarrow}{X}, \vec{X}) \circ (\overset{\leftarrow}{R}, \vec{R}) \subset (\overset{\leftarrow}{R}, \vec{R}) = (\overset{\leftarrow}{R}, \vec{R}) \circ (\overset{\leftarrow}{Q}, \vec{Q}) \circ (\overset{\leftarrow}{R}, \vec{R}) .$ $(\overset{\leftarrow}{R}, \vec{R}) \circ (\overset{\leftarrow}{X}, \vec{X}) \circ (\overset{\leftarrow}{R}, \vec{R}) \subset (\overset{\leftarrow}{R}, \vec{R}) = (\overset{\leftarrow}{R}, \vec{R}) \circ (\overset{\leftarrow}{Q}, \vec{Q}) \circ (\overset{\leftarrow}{R}, \vec{R}) .$

Thus, $(\overset{\leftarrow}{R}, \vec{R}) \circ (\overset{\leftarrow}{X}, \vec{X}) \circ (\overset{\leftarrow}{R}, \vec{R}) = (\overset{\leftarrow}{R}, \vec{R}),$ $(\overset{\leftarrow}{R}, \vec{R}) \circ (\overset{\leftarrow}{X}, \vec{X}) \circ (\overset{\leftarrow}{R}, \vec{R}) = (\overset{\leftarrow}{R}, \vec{R}),$ which completes the proof.

2. Dynamic fuzzy matrix HRL algorithm

Learning problems can be decomposed in accordance with their level so as to construct a bottom-up hierarchy. This is more conducive to qualitative and quantitative analysis. The traditional absolute value is used to show the uncertainty that does not accord with the subjective judgment and the characteristics of the object, so dynamic fuzzy theory is used to analyse the learning problem. We use the dynamic fuzzy matrix to represent the relationship rate between upper data and related data in this level. Assuming that upper element C and this level’s elements c1, c2, . . ., cn are related, then the dynamic fuzzy matrix $(\overset{\leftarrow}{R}, \vec{R})$ $(\overset{\leftarrow}{R}, \vec{R})$ has the following form:

$(\begin{array}{l} C & c_{1} & c_{2} & \dots & c_{n} \\ c_{1} & ({\overset{\leftarrow}{r}}_{11}, {\vec{r}}_{11}) & ({\overset{\leftarrow}{r}}_{12}, {\vec{r}}_{12}) & \dots & ({\overset{\leftarrow}{r}}_{18}, {\vec{r}}_{1 n}) \\ c_{2} & ({\overset{\leftarrow}{r}}_{21}, {\vec{r}}_{21}) & ({\overset{\leftarrow}{r}}_{22}, {\vec{r}}_{22}) & \dots & ({\overset{\leftarrow}{r}}_{28}, {\vec{r}}_{2 n}) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ c_{n} & ({\overset{\leftarrow}{r}}_{n 1}, {\vec{r}}_{n 1}) & ({\overset{\leftarrow}{r}}_{n 2}, {\vec{r}}_{n 2}) & \dots & ({\overset{\leftarrow}{r}}_{n n}, {\vec{r}}_{n m}) \end{array})$ $(\begin{array}{l} C & c_{1} & c_{2} & \dots & c_{n} \\ c_{1} & ({\overset{\leftarrow}{r}}_{11}, {\vec{r}}_{11}) & ({\overset{\leftarrow}{r}}_{12}, {\vec{r}}_{12}) & \dots & ({\overset{\leftarrow}{r}}_{18}, {\vec{r}}_{1 n}) \\ c_{2} & ({\overset{\leftarrow}{r}}_{21}, {\vec{r}}_{21}) & ({\overset{\leftarrow}{r}}_{22}, {\vec{r}}_{22}) & \dots & ({\overset{\leftarrow}{r}}_{28}, {\vec{r}}_{2 n}) \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋮ \\ c_{n} & ({\overset{\leftarrow}{r}}_{n 1}, {\vec{r}}_{n 1}) & ({\overset{\leftarrow}{r}}_{n 2}, {\vec{r}}_{n 2}) & \dots & ({\overset{\leftarrow}{r}}_{n n}, {\vec{r}}_{n m}) \end{array})$

where $({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j}) \in [0, 1] \times [\leftarrow, \to] (i, j \leq n)$ $({\overset{\leftarrow}{r}}_{i j}, {\vec{r}}_{i j}) \in [0, 1] \times [\leftarrow, \to] (i, j \leq n)$ represents element ci and cj according to the relationship rate of element C and elements ci and cj contain the membership rates of the dynamic fuzzy relationship. According to the relationship between elements c1, c2, . . ., cn and upper element C, we construct the dynamic fuzzy matrix of each level as

Definition 6.9 For a dynamic fuzzy number M in dynamic fuzzy matrix $(\overset{\leftarrow}{R}, \vec{R})$ $(\overset{\leftarrow}{R}, \vec{R})$ if and only if its membership rate ${({\overset{\leftarrow}{μ}}_{M}, {\vec{μ}}_{M})}_{(\overset{\leftarrow}{x}, \vec{x})} \in [0, 1] \times [\leftarrow, \to]$ ${({\overset{\leftarrow}{μ}}_{M}, {\vec{μ}}_{M})}_{(\overset{\leftarrow}{x}, \vec{x})} \in [0, 1] \times [\leftarrow, \to]$ satisfies

${({\overset{\leftarrow}{μ}}_{M}, {\vec{μ}}_{M})}_{(\overset{\leftarrow}{x}, \vec{x})} = {\begin{cases} \frac{(\overset{\leftarrow}{l} - \overset{\leftarrow}{x}, \vec{l} - \overset{\leftarrow}{x})}{(\overset{\leftarrow}{l} - \overset{\leftarrow}{m}, \overset{\leftarrow}{l} - \vec{m})}, (\overset{\leftarrow}{x}, \vec{x}) \in [l, m] \times [\leftarrow, \to] \\ \frac{(\overset{\leftarrow}{x} - \overset{\leftarrow}{n}, \vec{x} - \overset{\leftarrow}{n})}{(\overset{\leftarrow}{m} - \overset{\leftarrow}{n}, \overset{\leftarrow}{m} - \vec{n})}, (\overset{\leftarrow}{x}, \vec{x}) \in [m, n] \times [\leftarrow, \to] \\ 0, o t h e r w i s e \end{cases} (6.6)$ ${({\overset{\leftarrow}{μ}}_{M}, {\vec{μ}}_{M})}_{(\overset{\leftarrow}{x}, \vec{x})} = {\begin{cases} \frac{(\overset{\leftarrow}{l} - \overset{\leftarrow}{x}, \vec{l} - \overset{\leftarrow}{x})}{(\overset{\leftarrow}{l} - \overset{\leftarrow}{m}, \overset{\leftarrow}{l} - \vec{m})}, (\overset{\leftarrow}{x}, \vec{x}) \in [l, m] \times [\leftarrow, \to] \\ \frac{(\overset{\leftarrow}{x} - \overset{\leftarrow}{n}, \vec{x} - \overset{\leftarrow}{n})}{(\overset{\leftarrow}{m} - \overset{\leftarrow}{n}, \overset{\leftarrow}{m} - \vec{n})}, (\overset{\leftarrow}{x}, \vec{x}) \in [m, n] \times [\leftarrow, \to] \\ 0, o t h e r w i s e \end{cases} (6.6)$

then M is a triangular dynamic fuzzy number, written as $M = ((\overset{\leftarrow}{l}, \vec{l}), (\overset{\leftarrow}{m}, \vec{m}),$ $M = ((\overset{\leftarrow}{l}, \vec{l}), (\overset{\leftarrow}{m}, \vec{m}),$ $(\vec{n}, \vec{n})) . Here, (\overset{\leftarrow}{l}, \vec{l}) \leq (\overset{\leftarrow}{m}, \vec{m}) \leq (\overset{\leftarrow}{n}, \vec{n}),$ $(\vec{n}, \vec{n})) . Here, (\overset{\leftarrow}{l}, \vec{l}) \leq (\overset{\leftarrow}{m}, \vec{m}) \leq (\overset{\leftarrow}{n}, \vec{n}),$ and all M belong to [0, 1] × [←, →], where $(\overset{\leftarrow}{l}, \vec{l}) and (\overset{\leftarrow}{n}, \vec{n})$ $(\overset{\leftarrow}{l}, \vec{l}) and (\overset{\leftarrow}{n}, \vec{n})$ denote the lower and upper bounds of M, respectively.

Consider two groups of triangular dynamic fuzzy number $M_{1} = (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}),$ $M_{1} = (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}),$ $({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}), ({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1})) and M_{2} = (({\overset{↼}{l}}_{2}, {\overset{⇀}{l}}_{2}), ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) .$ $({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}), ({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1})) and M_{2} = (({\overset{↼}{l}}_{2}, {\overset{⇀}{l}}_{2}), ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) .$ Mathematical operations on these numbers are performed as follows:

$\begin{array}{l} (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}), ({\overset{\leftarrow}{m}}_{1}, \vec{m_{1}})), ({\overset{\leftarrow}{n}}_{1} {\vec{n}}_{1}) \oplus (({\overset{\leftarrow}{l}}_{2}, {\vec{l}}_{2}), ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) \\ = (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}) + ({\overset{\leftarrow}{l}}_{2}, {\vec{l}}_{2}), ({\overset{\leftarrow}{m}}_{1} {\vec{m}}_{1}) + ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1}) + ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) (6.7) \end{array}$ $\begin{array}{l} (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}), ({\overset{\leftarrow}{m}}_{1}, \vec{m_{1}})), ({\overset{\leftarrow}{n}}_{1} {\vec{n}}_{1}) \oplus (({\overset{\leftarrow}{l}}_{2}, {\vec{l}}_{2}), ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) \\ = (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}) + ({\overset{\leftarrow}{l}}_{2}, {\vec{l}}_{2}), ({\overset{\leftarrow}{m}}_{1} {\vec{m}}_{1}) + ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1}) + ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) (6.7) \end{array}$

$\begin{array}{l} (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}), ({\overset{\leftarrow}{m}}_{1}, \vec{m_{1}})), ({\overset{\leftarrow}{n}}_{1} {\vec{n}}_{1}) \otimes (({\overset{\leftarrow}{l}}_{2}, {\vec{l}}_{2}), ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) \\ \approx (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}) ({\overset{\leftarrow}{l}}_{2}, {\vec{l}}_{2}), ({\overset{\leftarrow}{m}}_{1} {\vec{m}}_{1}) ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1}) ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) (6.8) \end{array}$ $\begin{array}{l} (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}), ({\overset{\leftarrow}{m}}_{1}, \vec{m_{1}})), ({\overset{\leftarrow}{n}}_{1} {\vec{n}}_{1}) \otimes (({\overset{\leftarrow}{l}}_{2}, {\vec{l}}_{2}), ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) \\ \approx (({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}) ({\overset{\leftarrow}{l}}_{2}, {\vec{l}}_{2}), ({\overset{\leftarrow}{m}}_{1} {\vec{m}}_{1}) ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}), ({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1}) ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) (6.8) \end{array}$

${((\overset{\leftarrow}{l}, {\vec{l}}_{1}), ({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}), ({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1}))}^{- 1} \approx (\frac{1}{({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1})}, \frac{1}{({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1})}, \frac{1}{({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1})}) (6.9)$ ${((\overset{\leftarrow}{l}, {\vec{l}}_{1}), ({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}), ({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1}))}^{- 1} \approx (\frac{1}{({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1})}, \frac{1}{({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1})}, \frac{1}{({\overset{\leftarrow}{n}}_{1}, {\vec{n}}_{1})}) (6.9)$

Define the possibility rate of M1 ≥ M2, then

$V (M_{1} \geq M_{2}) = \sup_{(\overset{\leftarrow}{x}, \vec{x}) \geq (\overset{\leftarrow}{y}, \vec{y})} (\land ({({\overset{\leftarrow}{μ}}_{M_{1}}, {\vec{μ}}_{M_{1}})}_{(\overset{\leftarrow}{x}, \vec{x})}, ({\overset{\leftarrow}{μ}}_{M_{2}}, {\vec{μ}}_{M_{2}}) (\overset{\leftarrow}{y}, \vec{y}))) . (6.10)$ $V (M_{1} \geq M_{2}) = \sup_{(\overset{\leftarrow}{x}, \vec{x}) \geq (\overset{\leftarrow}{y}, \vec{y})} (\land ({({\overset{\leftarrow}{μ}}_{M_{1}}, {\vec{μ}}_{M_{1}})}_{(\overset{\leftarrow}{x}, \vec{x})}, ({\overset{\leftarrow}{μ}}_{M_{2}}, {\vec{μ}}_{M_{2}}) (\overset{\leftarrow}{y}, \vec{y}))) . (6.10)$

in which $({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}) \geq ({\vec{m}}_{2}, {\vec{m}}_{2})$ $({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}) \geq ({\vec{m}}_{2}, {\vec{m}}_{2})$ is the sufficient and necessary condition of V(M1 ≥ M2) = 1.

When $({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}) < ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}),$ $({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}) < ({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}),$ define the possibility of M1 ≥ M2 as follows:

$V (M_{1} \geq M_{2}) = h g t (M_{1} \cap M_{2}) = {(\overset{\leftarrow}{μ}, \vec{μ})}_{(\overset{\leftarrow}{d}, \vec{d})}, (6.11)$ $V (M_{1} \geq M_{2}) = h g t (M_{1} \cap M_{2}) = {(\overset{\leftarrow}{μ}, \vec{μ})}_{(\overset{\leftarrow}{d}, \vec{d})}, (6.11)$

where $(\overset{\leftarrow}{d}, \vec{d})$ $(\overset{\leftarrow}{d}, \vec{d})$ is the abscissa of the crossover between M1, M2, as shown in Fig. 6.2.

Fig. 6.2: Illustration for two triangular dynamic fuzzy numbers.

We can know,

$\begin{aligned} \frac{(\overset{\leftarrow}{l_{1}}, \vec{l_{1}}) - (\overset{\leftarrow}{n_{2}}, \vec{n_{2}})}{(({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}) - ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) - (({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}) - ({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}))}, \\ 0, o t h e r w i s e \end{aligned}$ $\begin{aligned} \frac{(\overset{\leftarrow}{l_{1}}, \vec{l_{1}}) - (\overset{\leftarrow}{n_{2}}, \vec{n_{2}})}{(({\overset{\leftarrow}{m}}_{2}, {\vec{m}}_{2}) - ({\overset{\leftarrow}{n}}_{2}, {\vec{n}}_{2})) - (({\overset{\leftarrow}{m}}_{1}, {\vec{m}}_{1}) - ({\overset{\leftarrow}{l}}_{1}, {\vec{l}}_{1}))}, \\ 0, o t h e r w i s e \end{aligned}$

Use $f = ((\overset{\leftarrow}{l}, \vec{l}), (\overset{\leftarrow}{m}, \vec{m}), (\overset{\leftarrow}{n}, \vec{n}))$ $f = ((\overset{\leftarrow}{l}, \vec{l}), (\overset{\leftarrow}{m}, \vec{m}), (\overset{\leftarrow}{n}, \vec{n}))$ converse dynamic fuzzy matrix into

$(\overset{\leftarrow}{R^{'}}, \vec{R^{'}}) = (\begin{matrix} (({\overset{\leftarrow}{r}}_{11 l}, {\vec{r}}_{11 l}), & (({\overset{\leftarrow}{r}}_{12 l}, {\vec{r}}_{12 l}), & ({\overset{\leftarrow}{r}}_{1 n l}, {\vec{r}}_{1 N l}), \\ ({\overset{\leftarrow}{r}}_{11 m}, {\vec{r}}_{11 m}), & ({\overset{\leftarrow}{r}}_{12 m}, {\vec{r}}_{12 m}), & \dots & ({\overset{\leftarrow}{r}}_{1 n m}, {\vec{r}}_{1 n m}), \\ ({\overset{\leftarrow}{r}}_{11 n}, {\vec{r}}_{11 n})) & ({\overset{\leftarrow}{r}}_{12 n}, {\vec{r}}_{12 n})) & ({\overset{\leftarrow}{r}}_{1 n n}, {\vec{r}}_{1 n N})) \\ (({\overset{\leftarrow}{r}}_{21 l}, {\vec{r}}_{21 l}), & (({\overset{\leftarrow}{r}}_{22 l}, {\vec{r}}_{22 l}), & ({\overset{\leftarrow}{r}}_{2 n l}, {\vec{r}}_{2 n l}), \\ ({\overset{\leftarrow}{r}}_{21 m}, {\vec{r}}_{21 m}), & ({\overset{\leftarrow}{r}}_{22 m}, {\vec{r}}_{22 m}), & ⋮ & ({\overset{\leftarrow}{r}}_{2 n m}, {\vec{r}}_{2 n m}), \\ ({\overset{\leftarrow}{r}}_{21 n}, {\vec{r}}_{21 n})) & ({\overset{\leftarrow}{r}}_{22 n}, {\vec{r}}_{22 n})) & ({\overset{\leftarrow}{r}}_{2 n n}, {\vec{r}}_{2 N n})) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ (({\overset{\leftarrow}{r}}_{N 1 l}, {\vec{r}}_{n 1 l}), & (({\overset{\leftarrow}{r}}_{n 2 l}, {\vec{r}}_{n 2 l}), & (({\overset{\leftarrow}{r}}_{N n l}, {\vec{r}}_{N n l}), \\ ({\overset{\leftarrow}{r}}_{n 1 m}, {\vec{r}}_{11 m}), & ({\overset{\leftarrow}{r}}_{n 2 m}, {\vec{r}}_{n 2 m}), & \dots & ({\overset{\leftarrow}{r}}_{n n m}, {\vec{r}}_{n n m}), \\ ({\overset{\leftarrow}{r}}_{n 1 n}, {\vec{r}}_{n 1 n})) & ({\overset{\leftarrow}{r}}_{n 2 n}, {\vec{r}}_{N 2 n})), & ({\overset{\leftarrow}{r}}_{n n n}, {\vec{r}}_{n n n})) \end{matrix}) .$ $(\overset{\leftarrow}{R^{'}}, \vec{R^{'}}) = (\begin{matrix} (({\overset{\leftarrow}{r}}_{11 l}, {\vec{r}}_{11 l}), & (({\overset{\leftarrow}{r}}_{12 l}, {\vec{r}}_{12 l}), & ({\overset{\leftarrow}{r}}_{1 n l}, {\vec{r}}_{1 N l}), \\ ({\overset{\leftarrow}{r}}_{11 m}, {\vec{r}}_{11 m}), & ({\overset{\leftarrow}{r}}_{12 m}, {\vec{r}}_{12 m}), & \dots & ({\overset{\leftarrow}{r}}_{1 n m}, {\vec{r}}_{1 n m}), \\ ({\overset{\leftarrow}{r}}_{11 n}, {\vec{r}}_{11 n})) & ({\overset{\leftarrow}{r}}_{12 n}, {\vec{r}}_{12 n})) & ({\overset{\leftarrow}{r}}_{1 n n}, {\vec{r}}_{1 n N})) \\ (({\overset{\leftarrow}{r}}_{21 l}, {\vec{r}}_{21 l}), & (({\overset{\leftarrow}{r}}_{22 l}, {\vec{r}}_{22 l}), & ({\overset{\leftarrow}{r}}_{2 n l}, {\vec{r}}_{2 n l}), \\ ({\overset{\leftarrow}{r}}_{21 m}, {\vec{r}}_{21 m}), & ({\overset{\leftarrow}{r}}_{22 m}, {\vec{r}}_{22 m}), & ⋮ & ({\overset{\leftarrow}{r}}_{2 n m}, {\vec{r}}_{2 n m}), \\ ({\overset{\leftarrow}{r}}_{21 n}, {\vec{r}}_{21 n})) & ({\overset{\leftarrow}{r}}_{22 n}, {\vec{r}}_{22 n})) & ({\overset{\leftarrow}{r}}_{2 n n}, {\vec{r}}_{2 N n})) \\ ⋮ & ⋮ & ⋮ & ⋮ \\ (({\overset{\leftarrow}{r}}_{N 1 l}, {\vec{r}}_{n 1 l}), & (({\overset{\leftarrow}{r}}_{n 2 l}, {\vec{r}}_{n 2 l}), & (({\overset{\leftarrow}{r}}_{N n l}, {\vec{r}}_{N n l}), \\ ({\overset{\leftarrow}{r}}_{n 1 m}, {\vec{r}}_{11 m}), & ({\overset{\leftarrow}{r}}_{n 2 m}, {\vec{r}}_{n 2 m}), & \dots & ({\overset{\leftarrow}{r}}_{n n m}, {\vec{r}}_{n n m}), \\ ({\overset{\leftarrow}{r}}_{n 1 n}, {\vec{r}}_{n 1 n})) & ({\overset{\leftarrow}{r}}_{n 2 n}, {\vec{r}}_{N 2 n})), & ({\overset{\leftarrow}{r}}_{n n n}, {\vec{r}}_{n n n})) \end{matrix}) .$

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6 Dynamic fuzzy hierarchical relationships