Some Linguistic Scaling Issues of Fuzzy Sets Based Decision-Making Systems

Alex Tserkovny

Applied AI Services, Brookline, USA.

DOI: 10.4236/jsea.2025.1811026                

Abstract

In this study we are examining expert assessment (EA) non-linear scaled based approach to map original physical scale into a linguistical one. We introduce fuzzification of linguistic clustering mechanism for both input and output of a model under investigation, followed by appropriate decision-making apparatus. We also offered defuzzification approach for presented fuzzy model output. All results are thoroughly illustrated by proper experimental results.

Keywords

Linguistic Scale, Cluster, Centroid, Fuzzy Logic, Modus Ponnens, Fuzzy Conditional Inference

1. Introduction

The original fuzzy clustering (also referred to as soft clustering or soft k-means) [1]-[6] is known as a form of clustering in which each data point can belong to more than one cluster. Usually, this technique involves assigning data points to clusters such that items in the same cluster are as similar as possible, while items belonging to different clusters are as dissimilar as possible. Clusters are identified via similarity measures, which include distance, connectivity, and intensity. Different similarity measures may be chosen based on the data or the application.

In this study we are investigating very different dimensions of fuzzy clustering, related to linguistic determination of each cluster by EAs, associated with corresponding physical values. Note, that in all our previous studies [7]-[13] we always suggested that there is linear (monotonic) mapping between considered physical scale and its linguistic counterpart. It always meant, that if you have a scale of certain physical component $X$ , situated between its marginal $\left[X^{\min },X^{\max }\right]$ values, then it is very “obvious” to consider dividing this scale by equal intervals with further assignment of certain linguistic label to each of them. Thus, if there is N number of linguistic labels, then the way to map them into a scale of $X$ is to simply divide its length by ($N-1$ ), i.e. $\Delta X=\frac{X}{N-1}$ and assign each element of this set $S_{ }=\left\{j\cdot \Delta X\right\}|\forall j\in \left[0,N-1\right]$ to its linguistic counterpart (label) from $L=\left\{l_j\right\}|\forall j\in \left[0,N-1\right]$ . Such linguistic approach permitted to use simple triangular shaped membership functions (MF).

In this investigation, in contrast, we use a different approach, based on EAs, which allows to assign linguistic labels not to evenly divided physical scale, but rather to arbitrary intervals within it, associated with expert’s knowledge and considerations about a nature of represented physical component. Thus, considered physical scale would be divided by certain number of linguistically labeled clusts. This ah makes fuzzy decision-making mechanism to be more properly applied to a nature of real-world behavior. In this study we cover issues of scaling, building fuzzy clusters, labeling, fuzzification of a model input/output, aggregation of fuzzy elements of a model and appropriate fuzzy decision-making mechanism. We also present corresponding defuzzification mechanism.

2. Scaling and Clustering

2.1. Scales

Let us consider the fact that both input $x$ and output $y$ values of our model situate within certain min/max boundaries (in some abstract physical units), called scales.

$x\in \left[x^{\min },x^{\max }\right],y\in \left[y^{\min },y^{\max }\right]$

We also suggest that both scales could consist of certain number of sub-scales $x_i|\forall i\in \left[0,N\right]$ and $y_m|\forall m\in \left[0,N\right]$ , each of which represents some semantic values, proposed by certain EA, made based on an expert knowledge about physical nature of a scale. (For simplification’s sake we suppose the same number of sub-scales N for both input and output). Above mentioned semantic values must be presented for experts as a set of linguistic terms, associated with corresponding sub-scales, i.e. intervals of certain physical values. These terms usually form linguistic scales such as [“ow”…“medium”…“high”], [“small”…“average”…“large”] or [“near”…“in between”…“far”] etc.

$x_i\in \left[x_i^{\min },x_i^{\max }\right]|\forall i\in \left[0,N\right]$ , $y_m\in \left[y_m^{\min },y_m^{\max }\right]|\forall m\in \left[0,N\right]$ , (2.1)

The following feature for both input and output scales is taking place, i.e. the min value of each subsequent sub-scale is equal to max value of its predecessor.

$x_i^{\max }=x_{i+1}^{\min }|\forall i\in \left[0,N-1\right]$ , $y_m^{\max }=y_{m+1}^{\min }|\forall m\in \left[0,N-1\right]$

Thus, both input and output scales would be represented in terms of their correspondent sub-scales

$x=\left\{\left[x_0^{\min },x_0^{\max }\right],\left[x_1^{\min },x_1^{\max }\right],\cdots ,\left[x_N^{\min },x_N^{\max }\right]\right\}$ ,

$y=\left\{\left[y_0^{\min },y_0^{\max }\right],\left[y_1^{\min },y_1^{\max }\right],\cdots ,\left[y_N^{\min },y_N^{\max }\right]\right\}$ .

We could estimate the length of each sub-scale for the case when the original scale divided equally.

For an input we have

$x_{even}=\frac{x^{\max }-x^{\min }}{N}$ (2.2)

For an output we have

$y_{even}=\frac{y^{\max }-y^{\min }}{N}$ (2.3)

From (2.2) and (2.3) we define the relative weight of each sub-scale length the following way.

${\alpha }_i=\frac{x_i^{\max }-x_i^{\min }}{x_{even}}|\forall i\in \left[0,N\right]$ ,(2.4)

${\beta }_m=\frac{y_m^{\max }-y_m^{\min }}{y_{even}}|\forall m\in \left[0,N\right]$ (2.5)

Note that both ${\alpha }_i|\forall i\in \left[0,N\right]$ and ${\beta }_m|\forall m\in \left[0,N\right]$ sets define the relative value of each sub-scale involvement in decision making process, described down below. The larger this relative value, the lower the degree of its importance and vice versa. By using (2.4) and (2.5) we introduce the minimal value of relative weights over all sub-scales as the following

${\alpha }_{\min }=\underset{i\in \left[0,N\right]}{\min }\left\{{\alpha }_i\right\}$ (2.6)

${\beta }_{\min }=\underset{m\in \left[0,N\right]}{\min }\left\{{\beta }_m\right\}$ (2.7)

Both ${\alpha }_{\min }$ and ${\beta }_{\min }$ will define the shape of membership functions (MF) for both input and output of a model, represented down below as corresponding fuzzy sets.

2.2. Clusters

We redefine each sub-scale $x_i|\forall i\in \left[0,N\right]$ and $y_m|\forall m\in \left[0,N\right]$ , as a set of clusters, each of which could be characterized by its median value or so-called a centroid of the following type.

$ic{l}_i=\frac{x_i^{\max }+x_i^{\min }}{2}|\forall i\in \left[0,N\right]$ , $oc{l}_m=\frac{y_m^{\max }+y_m^{\min }}{2}|\forall m\in \left[0,N\right]$ (2.8)

2.3. Generated Experimental Data

To sustain objectivity of our experiments we use the generator of random numbers to produce experimental data. First, we created the following “physical” scales (in abstract unit values).

Input Scale: [0, 12960], Output Scale: [0, 108].

We will use the number of sub-scales/clusters N = 8.

Then from (2.1) and (2.2) we are getting $x_{\text{even}}=1440.0$ , $y_{\text{even}}=12.0$

By using a random number generator, we found the following sub-scales/clusters for an input.

$x_0\in \left[0,1440\right]$ ; $x_1\in \left[1440,1800\right]$ ; $x_2\in \left[1800,4320\right]$ ; $x_3\in \left[4320,7200\right]$ ; $x_4\in \left[7200,9000\right]$ ; $x_5\in \left[9000,11160\right]$ ; $x_6\in \left[11160,12240\right]$ ; $x_7\in \left[12240,12960\right]$ .

For each input sub-scale/cluster we define an abstract simplified linguistic term from the following set $T\left(x\right)$ = {“in0”, “in1”, “in2”, “in3”, “in4”, “in5”, “in6”, “in7”}.

We also found the following sub-scales for an output.

$y_0\in \left[0,21\right]$ ; $y_1\in \left[21,36\right]$ ; $y_2\in \left[36,60\right]$ ; $y_3\in \left[60,63\right]$ ; $y_4\in \left[63,69\right]$ ; $y_5\in \left[69,87\right]$ ; $y_6\in \left[87,99\right]$ ; $y_7\in \left[99,108\right]$ .

By analogy we define linguistic terms for each output sub-scale/cluster, i.e. $T\left(y\right)$ = {“out0”, “ out1”, “out2”, “out3”, “out4”, “out5”, “out6”, “out7”}.

Following Table 1 depicts corresponding results.

Table 1. Linguistic terms for input/output clusters.

Value of variable		$ic{l}_i\in U,i=\overline{0,7}$ $oc{l}_j\in V,j=\overline{0,7}$
ICL	OCL
in0	out0	0
in1	out1	1
in2	out2	2
in3	out3	3
in4	out4	4
in5	out5	5
in6	out6	6
in7	out7	7

From (2.8) by using $x_i|\forall i\in \left[0,7\right]$ and $y_m|\forall m\in \left[0,7\right]$ we are getting two sets of centroids ICL = {720, 1620, 3060, 5760, 8100, 10080, 11700, 12600} – for the input and OCL = {10, 28, 48, 61, 66, 78, 93, 103} – for the output.

By using (2.4) and (2.5) we are getting values for ${\alpha }_i|\forall i\in \left[0,N\right]$ , and ${\beta }_i|\forall m\in \left[0,N\right]$ .

Following Table 2 depicts corresponding results.

Table 2. Experimental data.

Input Term	$x^{min}$	ICL centroid	$x^{max}$	$\alpha$	Output Term	$y^{min}$	OCL centroid	$y^{max}$	$\beta$
“in0”	0	720	1440	1.0	“out0”	0	10	21	1.75
“in1”	1440	1620	1800	0.25	“out1”	21	28	36	1.25
“in2”	1800	3060	4320	1.75	“out2”	36	48	60	2.0
“in3”	4320	5760	7200	2.0	“out3”	60	61	63	0.25
“in4”	7200	8100	9000	1.25	“out4”	63	66	69	0.5
“in5”	9000	10080	11160	1.5	“out5”	69	78	87	1.5
“in6”	11160	11700	12240	0.75	“out6”	87	93	99	1.0
“in7”	12240	12600	12960	0.5	“out7”	99	103	108	0.75

3. Fuzzy Reasoning

3.1. Fuzzification of Input/Output Centroids

We represent each input centroid of i-th cluster $ic{l}_i|\forall i\in \left[0,N\right]$ as a fuzzy set, forming linguistic variable, described by a triplet of the form

$IC{L}_i=\left\{\langle ic{l}_i,U,\widetilde{IC{L}_i}\rangle \right\},ic{l}_i\in T\left(u\right)$ , $\forall i\in \left[0,N\right]$ ,

where U is a universe of discourse of a model input, $T\left(u\right)$ is extended term set of the linguistic variable “ICL” from Table 1,

$T\left(u\right)=\left\{t_0^{in},t_1^{in},t_2^{in},\cdots ,t_N^{in}\right\}.$

$\widetilde{IC{L}_i}|\forall i\in \left[0,N\right]$ is normal fuzzy set with correspondent MF ${\mu }_{ic{l}_i}:U\to \left[0,1\right]|\forall i\in \left[0,N\right]$ . To normalize values of $ic{l}_i|\forall i\in \left[0,N\right]$ we use the following

$ic{l}_i^{norm}=\frac{ic{l}_i-ic{l}_i^{\min }}{ic{l}_i^{\max }-ic{l}_i^{\min }{}_{ }},$ (3.1)

We will use the following mapping $\partial :\widetilde{IC{L}_i}\to U|u=Ent\left[\left(CardU-1\right)\cdot ic{l}_i^{norm}\right]$ , where

$\widetilde{IC{L}_i}={\displaystyle {\int }_U{\mu }_{ic{l}_i}\left(u\right)/u}$ , $\forall i\in \left[0,N\right]$ (3.2)

On the other hand, to determine the estimates of the MF in terms of singletons from (3.2) in the form ${\mu }_{ic{l}_i}\left(u_j\right)/u_j|\forall j\in \left[0,CardU\right],\forall i\in \left[0,N\right]$ , given (3.1) we propose the following procedure.

$\begin{array}{l}{\mu }_{ic{l}_i}\left(u_j\right)={\left\{1-\frac{1}{CardU-1}\cdot \left|j-Ent\left[\left(CardU-1\right)\cdot ic{l}_i^{norm}\right]\right|\right\}}^{{\alpha }_i},\\ \forall j\in \left[0,CardU\right],\forall i\in \left[0,N\right]\end{array}$ (3.3)

where ${\alpha }_i$ is a weight coefficient of the i-th input cluster from (2.4). Notice that we define all values of a linguistic variable over entire physical scale of input/output parameters via normalization mechanism and therefore mathematically reject the notion of interval based MFs.

Each output centroid of a cluster $oc{l}_m|\forall m\in \left[0,N\right]$ we represent as a fuzzy set, forming linguistic variable, described by a triplet of the form

$OC{L}_m=\left\{\langle oc{l}_m,V,\widetilde{OC{L}_m}\rangle \right\},oc{l}_m\in T\left(v\right),\forall m\in \left[0,N\right]$ ,

where V is a universe of discourse of a model output, $T\left(v\right)$ is extended term set of the linguistic variable “OCL” from Table 1,

$T\left(v\right)=\left\{t_0^{out},t_1^{out},t_2^{out},\cdots ,t_N^{out}\right\}.$

$\widetilde{OCL}$ is nomal fuzzy set with correspondent MF ${\mu }_{oc{l}_m}:V\to \left[0,1\right]|\forall m\in \left[0,N\right]$ . We use the same normalization procedure

$oc{l}_m^{norm}=\frac{oc{l}_m-oc{l}_m^{\min }}{oc{l}_m^{\max }-oc{l}_m^{\min }},\forall m\in \left[0,N\right]$ (3.4)

With the following mapping $\Omega :\widetilde{OC{L}_m}\to V|v=Ent\left[\left(CardV-1\right)\cdot oc{l}_m^{norm}\right]$ , where

$\widetilde{OC{L}_m}={\displaystyle {\int }_V{\mu }_{oc{l}_m}\left(v\right)/v},\forall m\in \left[0,N\right]$ (3.5)

On the other hand, similarly to the previous cases, to determine the estimates of the MF in terms of singletons from (3.5) in the form ${\mu }_{oc{l}_m}\left(v_k\right)/v_k|\forall k\in \left[0,CardV\right],\forall m\in \left[0,N\right]$ , given (3.4) we propose the following procedure.

$\begin{array}{l}{\mu }_{oc{l}_m}\left(v_k\right)={\left\{1-\frac{1}{CardV-1}\cdot \left|k-Ent\left[\left(CardV-1\right)\cdot oc{l}_m^{norm}\right]\right|\right\}}^{{\beta }_m}\\ \forall k\in \left[0,Card{V}_{ }\right],\forall m\in \left[0,N\right]\end{array}$ (3.6)

where ${\beta }_m$ is a weight coefficient of m-th output cluster from (2.5). Note that the use of both ${\alpha }_i$ in (3.3) and ${\beta }_m$ in (3.6) changes a shape of MF from ordinary triangular form into variety of shapes, which could adequately map each cluster into aggregation form, described down below.

3.2. MFs of Input/Output Centroids

Using data from Table 1 and Table 2, given (3.3) we have the following MFs for input centroids

${\mu }_{icl}$ (“in0”) = 1.00/0 + 0.88/1 + 0.75/2 + 0.63/3 + 0.50/4 + 0.38/5 + 0.25/6 + 0.13/7

${\mu }_{icl}$ (“in1”) = 0.97/0 + 1.00/1 + 0.97/2 + 0.93/3 + 0.89/4 + 0.84/5 + 0.78/6 + 0.71/7

${\mu }_{icl}$ (“in2”) = 0.79/0 + 1.00/1 + 0.79/2 + 0.60/3 + 0.44/4 + 0.30/5 + 0.18/6 + 0.09/7

${\mu }_{icl}$ (“in3”) = 0.39/0 + 0.56/1 + 0.77/2 + 1.00/3 + 0.77/4 + 0.56/5 + 0.39/6 + 0.25/7

${\mu }_{icl}$ (“in4”) = 0.29/0 + 0.42/1 + 0.56/2 + 0.70/3 + 0.85/4 + 1.00/5 + 0.85/6 + 0.70/7

${\mu }_{icl}$ (“in5”) = 0.13/0 + 0.23/1 + 0.35/2 + 0.49/3 + 0.65/4 + 0.82/5 + 1.00/6 + 0.82/7

${\mu }_{icl}$ (“in6”) = 0.21/0 + 0.35/1 + 0.48/2 + 0.59/3 + 0.70/4 + 0.81/5 + 0.90/6 + 1.00/7

${\mu }_{icl}$ (“in7”) = 0.35/0 + 0.50/1 + 0.61/2 + 0.71/3 + 0.79/4 + 0.87/5 + 0.94/6 + 1.00/7

Here is correspondent Figure 1 depiction

Figure 1. MFs of input centroids.

Using data from Table 1 and Table 2, given (3.6) we have the following MFs for output centroids

${\mu }_{ocl}$ (“out0”) = 1.00/0 + 0.79/1 + 0.60/2 + 0.44/3 + 0.30/4 + 0.18/5 + 0.09/6 +0.03/7

${\mu }_{ocl}$ (“out1”) = 0.70/0 + 0.85/1 + 1.00/2 + 0.85/3 + 0.70/4 + 0.56/5 + 0.42/6 + .29/7

${\mu }_{ocl}$ (“out2”) = 0.39/0 + 0.56/1 + 0.77/2 + 1.00/3 + 0.77/4 + 0.56/5 + 0.39/6 + 0.25/7

${\mu }_{ocl}$ (“out3”) = 0.84/0 + 0.89/1 + 0.93/2 + 0.97/3 + 1.00/4 + 0.97/5 + 0.93/6 + 0.89/7

${\mu }_{ocl}$ (“out4”) = 0.71/0 + 0.79/1 + 0.87/2 + 0.94/3 + 1.00/4 + 0.94/5 + 0.87/6 + 0.79/7

${\mu }_{ocl}$ (“out5”) = 0.23/0 + 0.35/1 + 0.49/2 + 0.65/3 + 0.82/4 + 1.00/5 + 0.82/6 + 0.65/7

${\mu }_{ocl}$ (“out6”) = 0.25/0 + 0.38/1 + 0.50/2 + 0.63/3 + 0.75/4 + 0.88/5 + 1.00/6 + 0.88/7

${\mu }_{ocl}$ (“out7”) = 0.21/0 + 0.35/1 + 0.48/2 + 0.59/3 + 0.70/4 + 0.81/5 + 0.90/6 + 1.00/7

Here is correspondent Figure 2 depiction

Figure 2. MFs of output centroids.

3.3. Fuzzification of Input/Output

We represent each input value $x$ as a fuzzy set, forming linguistic variable, described by a triplet of the form

$X=\left\{\langle x,U,\tilde{X}\rangle \right\},x\in T\left(u\right)$ ,

where $\tilde{X}$ is normal fuzzy set with correspondent MF ${\mu }_X:U\to \left[0,1\right]$ . $T\left(u\right)$ is extended term set of the linguistic variable “Aggregate Input”, which corresponds to fuzzy set (3.16), described down below. Each term of the linguistic variable “Aggregate Input” would be the term from Table 1. for variable “ICL” or combination of them. To normalize values of $x$ we use the following

$x^{norm}=\frac{x-x^{\min }}{x^{\max }-x^{\min }},$ (3.7)

We will use the following mapping $\gamma :\tilde{X}\to U|u=Ent\left[\left(CardU-1\right)\cdot x^{norm}\right]$ , where

$\tilde{X}={\displaystyle {\int }_U{\mu }_x\left(u\right)/u}$ ,(3.8)

On the other hand, to determine the estimates of the MF in terms of singletons ${\mu }_X\left(u_j\right)/u_j|\forall j\in \left[0,CardU\right]$ from (3.8), given (3.7) we propose the following procedure.

${\mu }_X\left(u_j\right)={\left\{1-\frac{1}{CardU-1}\cdot \left|j-Ent\left[\left(CardU-1\right)\cdot x^{norm}\right]\right|\right\}}^{{}_{{\alpha }_{\min }}},\forall j\in \left[0,CardU\right],$ (3.9)

where ${\alpha }_{\min }$ is a smallest weight coefficient of all input clusters from (2.6). The use of ${\alpha }_{\min }$ in (3.9) changes a shape of original MF (${\alpha }_{\min }=1$ ) from usual triangular form into parabola one, which guarantees more uniform value distributions within set of singletons.

Similarly, we represent each output value $y_{ }$ as a fuzzy set, forming linguistic variable, described by a triplet of the form

$Y=\left\{\langle y,V,\tilde{Y}\rangle \right\},y\in T\left(v\right)$ , ${\mu }_Y:V\to \left[0,1\right]$

where $\tilde{Y}$ is normal fuzzy set with correspondent MF ${\mu }_Y:V\to \left[0,1\right]$ . $T\left(v\right)$ is extended term set of the linguistic variable “Aggregate Output”, which corresponds to fuzzy set (3.23), described down below. Each term of the linguistic variable “Aggregate Output” would be the term from Table 1. for variable “OCL” or combination of them. To normalize values of $y$ we use the following

$y^{norm}=\frac{y-y^{\min }}{y^{\max }-y^{\min }}.$ (3.10)

We will use the following mapping $\omega :\tilde{Y}\to V|v=Ent\left[\left(CardV-1\right)\cdot y^{norm}\right]$ , where

$\tilde{Y}={\displaystyle {\int }_V{\mu }_Y\left(v\right)/v}$ , (3.11)

And again, to determine the estimates of the MF in terms of singletons ${\mu }_Y\left(u_k\right)/u_k|\forall k\in \left[0,CardV\right]$ from (3.11), given (3.10) we propose the following procedure.

${\mu }_Y\left(u_k\right)={\left\{1-\frac{1}{CardV-1}\cdot \left|k-Ent\left[\left(CardV-1\right)\cdot y^{norm}\right]\right|\right\}}^{{}_{{\beta }_{\min }}},\forall k\in \left[0,CardV\right]$ (3.12)

3.4. Aggregation of Fuzzy Input

The aggregation procedure consists of two steps.

1) On the first step we define the set of fuzzy sets as intersections ${\tilde{X}}_i^{inter}$ of a current fuzzy set of input $\tilde{X}$ with each fuzzy set of input centroid $\widetilde{IC{L}_i}|\forall i\in \left[0,N\right]$ .

Thus, we define the following

${\tilde{X}}_i^{inter}=\tilde{X}\cap \widetilde{IC{L}_i},\forall i\in \left[0,N\right].$ (3.13)

The set of fuzzy sets ${\tilde{X}}_i^{inter}$ is presented in familiar way as

${\tilde{X}}_i^{inter}={\displaystyle {\int }_U{\mu }_{X_i^{inter}}\left(u\right)/u}|\forall i\in \left[0,N\right],$ (3.14)

And we determine correspondent MFs from (3.14) in terms of singletons as ${\mu }_{X_i^{inter}}\left(u_j\right)/u_j|\forall i\in \left[0,CardU\right],\forall j\in \left[0,CardU\right]$

2) On the second step we define a fuzzy set ${\tilde{X}}_{ }^{aggr}$ as a union of all fuzzy sets ${\tilde{X}}_i^{inter}$ , i.e.

${\tilde{X}}^{aggr}={\displaystyle {\cup }_{i=0}^N{\tilde{X}}_i^{inter}}$ , ${\tilde{X}}_i^{inter}\subseteq U$ , (3.15)

We also present the fuzzy sets ${\tilde{X}}_{ }^{aggr}$ as follows

${\tilde{X}}^{aggr}={\displaystyle {\int }_U{\mu }_{X^{aggr}}\left(u\right)/u},$ (3.16)

And we determine correspondent MF from (3.16) in terms of singletons as${\mu }_{X^{aggr}}\left(u_j\right)/u_j|\forall j\in \left[0,CardU\right]$

3.5. Fuzzy Inference

To convert (3.17) - (3.18) into fuzzy logic-based statement and terms from Table 1. we use a Fuzzy Conditional Inference Rule (FCIR), formulated by means of “common sense” as a following conditional clause:

P = “IF (x is X), THEN (y is Y)” (3.17)

In other words, we use fuzzy conditional inference of the following type [8]:

Ant 1: If Input is X, then Output is Y

Ant 2: Input is $X^{\prime }$

Cons: Output is $Y^{\prime }$ (3.18)

where $X,X^{\prime }\subseteq U$ and $Y,Y^{\prime }\subseteq V$ .

Note that statements (3.17) and (3.18) represent “modus-ponens” syllogism. Given that we use the following type of implication [1]

$X\to Y=\{\begin{array}{l}\left(1-x\right)\cdot y,x>y,\\ 1,x\le y\end{array}$ (3.19)

For practical purposes, described down below, we will use Fuzzy Conditional Rule (FCR) [14] of the following type

$\begin{array}{l}R\left(A_1\left(x\right),A_2\left(y\right)\right)=\left(X\times U\to V\times Y\right)\cap \left(\neg X\times U\to V\times \neg Y\right)\\ ={\displaystyle {\int }_{U\times V}\left({\mu }_x\left(u\right)\to {\mu }_y\left(v\right)\right)}\wedge \left(\left(1-{\mu }_x\left(u\right)\right)\to \left(1-{\mu }_y\left(v\right)\right)\right)/\left(u,v\right)\end{array}$ (3.20)

Given (3.19) from (3.20) we are getting

$\begin{array}{c}R\left(A_1\left(x\right),A_2\left(y\right)\right)=\left({\mu }_x\left(u\right)\to {\mu }_y\left(v\right)\right)\wedge \left(\left(1-{\mu }_x\left(u\right)\right)\to \left(1-{\mu }_y\left(v\right)\right)\right)\\ =\{\begin{array}{l}\left(1-{\mu }_x\left(u\right)\right)\cdot {\mu }_y\left(v\right),{\mu }_x\left(u\right)<{\mu }_y\left(v\right),\\ 1,{\mu }_x\left(u\right)={\mu }_y\left(v\right),\\ \left(1-{\mu }_y\left(v\right)\right)\cdot {\mu }_x\left(u\right),{\mu }_y\left(v\right)<{\mu }_x\left(u\right).\end{array}\end{array}$ (3.21)

Given a unary relationship $R\left(A_1\left(x^{\prime }\right)\right)={\tilde{X}}^{aggr}$ one can obtain the consequence $R\left(A_2\left(y^{\prime }\right)\right)$ by CRI to $R\left(A_1\left(x^{\prime }\right)\right)$ and $R\left(A_1\left(x\right),A_2\left(y\right)\right)$ of type (3.21):

$\begin{array}{l}R\left(A_2\left(y^{\prime }\right)\right)={\tilde{X}}^{aggr}\circ R\left(A_1\left(x\right),A_2\left(y\right)\right)\\ ={\displaystyle {\int }_U{\mu }_{X^{aggr}}\left(u\right)/u}\circ {\displaystyle {\int }_{U\times V}\left({\mu }_x\left(u\right)\to {\mu }_y\left(v\right)\right)}\wedge \left(\left(1-{\mu }_x\left(u\right)\right)\to \left(1-{\mu }_y\left(v\right)\right)\right)/\left(u,v\right)\\ ={\displaystyle {\int }_V{\displaystyle {\cup }_{X^{aggr}\in U}{\mu }_{X^{aggr}}\left(u\right)}}\wedge \left({\mu }_x\left(u\right)\to {\mu }_y\left(v\right)\right)\wedge \left(\left(1-{\mu }_x\left(u\right)\right)\to \left(1-{\mu }_z\left(v\right)\right)\right)]/v\end{array}$ (3.22)

The value of an output fuzzy set $\widetilde{Y^{\prime }}$ could be consider as

$\widetilde{Y^{\prime }}=R\left(A_2\left(y^{\prime }\right)\right)$ (3.23)

3.6. Aggregation of Fuzzy Output

The aggregation procedure consists of two steps.

1) On the first step we define the set of fuzzy sets as intersections ${\tilde{Y}}_j^{inter}$ of a current fuzzy set of input $\tilde{Y}$ with each fuzzy set of input centroid $\widetilde{OC{L}_j}|\forall j\in \left[0,N\right]$ .

Thus, we define the following

${\tilde{Y}}_j^{inter}=\widetilde{Y^{\prime }}\cap \widetilde{OC{L}_j},\forall j\in \left[0,N\right]$ (3.24)

The set of fuzzy sets ${\tilde{Y}}_j^{inter}$ is presented in familiar way as

${\tilde{Y}}_j^{inter}={\displaystyle {\int }_V{\mu }_{Y_j^{inter}}\left(v\right)/v}|\forall j\in \left[0,N\right],$ (3.25)

And we determine correspondent MFs from (3.14) in terms of singletons as ${\mu }_{Y_j^{inter}{}_{ }}\left(v_k{}_{ }\right)/v_k{}_{ }|\forall j\in \left[0,CardV\right],\forall k\in \left[0,CardV\right]$

2) On the second step we define a fuzzy set ${\tilde{Y}}^{aggr}$ as a union of all fuzzy sets ${\tilde{Y}}_j^{inter}$ , i.e.

${\tilde{Y}}^{aggr}={\displaystyle {\cup }_{j=0}^N{\tilde{Y}}_j^{inter}}$ , ${\tilde{Y}}_j^{inter}\subseteq V$ , $\forall j\in \left[0,N\right]$ (3.26)

We also present the fuzzy sets ${\tilde{Y}}^{aggr}$ as follows

${\tilde{Y}}^{aggr}={\displaystyle {\int }_U{\mu }_{X^{aggr}}\left(u\right)/u}$ (3.27)

3.7. Linguistic Classification and Defuzzification of Aggregated of Fuzzy Sets

3.7.1. Linguistic Classification

It is well known that any fuzzy set from (3.11) ${\tilde{P}}_{ }={\displaystyle {\int }_W{\mu }_P\left(w\right)/w}$ could be represented as the following sum of singletons

$\begin{array}{l}\tilde{P}={\displaystyle {\sum }_{i=\overline{0,CardW}}{\mu }_P\left(w_i\right)/w_i}\\ \Rightarrow \exists W_1\subseteq W|Card{W}_1<CardW;\forall k\in \left[0,Card{W}_1\right]\\ \Rightarrow {\mu }_P\left(w_k\right)/w_k=\underset{i=\overline{0,CardW}}{\max }\left\{{\mu }_P\left(w_i\right)/w_i\right\}\end{array}$ (3.28)

There are two cases of MF in use.

1) Case of unimodal MF (semantical non ambiguity) for fuzzy set $\tilde{P}$ , i.e. $Card{W}_1=1$ $\Rightarrow$ $\exists k^{*}!\in W_1|{\mu }_P\left(w_{k^{*}}\right)/w_{k^{*}}=\underset{i=\overline{0,CardW}}{\max }\left\{{\mu }_P\left(w_i\right)/w_i\right\}$ $\Rightarrow p\in T\left(w\right)=t_{k^{*}}^{out}$ . Thus, there is unique linguistic term $t_{k^{*}}^{out}$ , associated with fuzzy set $\tilde{P}$ . $Card{W}_1=1$ .

2) Case of polymodal MF (semantical ambiguity) for fuzzy set $\tilde{P}$ , i.e. $Card{W}_1>1$ $\Rightarrow$ $\exists i^{*},j^{*},\cdots !\in W_1|{\mu }_P\left(w_{i^{*}}\right)/w_{i^{*}}={\mu }_P\left(w_{j^{*}}\right)/w_{j^{*}}=\cdots =\underset{k=\overline{0,CardW}}{\max }\left\{{\mu }_P\left(w_k\right)/w_k\right\}$ ,$\forall k\in \left[0,CardW\right]$ $\Rightarrow$ $p\in T\left(w\right)=t_{i^{*}}^{out},t_{j^{*}}^{out},\cdots$ . Thus, there are few linguistic terms $t_{i^{*}}^{out},t_{j^{*}}^{out},\cdots$ , associated with fuzzy set $\tilde{Y}$ .

3.7.2. Defuzzification

From (2.8) we have the following defuzzied values for an input

$x_{Def}=\frac{1}{Card{W}_1}{\displaystyle {\sum }_{\forall k\in W_1}ic{l}_k}$ , (3.29)

And for output

$y_{Def}=\frac{1}{Card{W}_1}{\displaystyle {\sum }_{\forall k\in W_1}oc{l}_k}$ (3.30)

3.8. Formalization of Knowledge Base

In [7]-[13] we always consider the fact that for FCR (3.20) its fuzzy binary relationship matrix must be formulated by conditional clause, which contain terms, characterized by margins of physical scales in use. It means, that semantic knowledge about a system input/output relationship must be formulated as one of the following clauses (3.17):

“IF (x is “lowest”), THEN (y is “highest”)”

“IF (x is “lowest”), THEN (y is “lowest”)”

“IF (x is “highest”), THEN (y is “lowest”)”

“IF (x is “highest”), THEN (y is “highest”)”

In our case “lowest”/“highest” pair for an input is “in0”/“in7” and “out0”/“out7”—for an output similarly.

Such an approach guarantees logical non-contradiction of FCR (3.21), i.e. appropriate binary relationship matrix has one of major diagonals all of singles [15]. To achieve this goal, we use the following triangular shape MF

${\mu }_X\left(t_{k^{*}}^{in}\right)=\left\{1-\frac{1}{CardU-1}\cdot \left|j-k^{*}\right|\right\},\forall j\in \left[0,CardU\right],\exists k^{*}!\in \left[0,CardU\right]$ (3.31)

${\mu }_Y\left(t_{m^{*}}^{out}\right)=\left\{1-\frac{1}{CardV-1}\cdot \left|j-m^{*}\right|\right\},\forall j\in \left[0,CardV\right],\exists m^{*}!\in \left[0,CardV\right]$ (3.32)

In (3.31) we use term for $k^{*}=0$ |$t_0^{in}$ = “in0” and in (3.32) we use term for $m^{*}=7$ |$t_7^{out}$ = “out7”. Here are correspondent MFs

${\mu }_X$ (“in0”) = 1.00/0 + 0.88/1 + 0.75/2 + 0.63/3 + 0.50/4 + 0.38/5 + 0.25/6 + 0.13/7

${\mu }_Y$ (“out7”) = 0.13/0 + 0.25/1 + 0.38/2 + 0.50/3 + 0.63/4 + 0.75/5 + 0.88/6 + 1.00/7

Thus, as a result we will use the following binary relationship matrix

$R_1\left(A_1\left(x\right),A_2\left(y\right)\right)=\left({\mu }_X\left(‘‘\text{in}0’’\right)\to {\mu }_Y\left(‘‘\text{out}7’’\right)\right)$ . (3.33)

Applying (3.21) and (3.33) we get matrix from Table 3.

Table 3. Binary relationship matrix.

X◊Y	0	1	2	3	4	5	6	7
0	0.00	0.02	0.03	0.05	0.06	0.08	0.09	1.00
1	0.00	0.03	0.06	0.09	0.13	0.16	1.00	0.09
2	0.00	0.05	0.09	0.14	0.19	1.00	0.16	0.08
3	0.00	0.06	0.13	0.19	1.00	0.19	0.13	0.06
4	0.00	0.08	0.16	1.00	0.19	0.14	0.09	0.05
5	0.00	0.09	1.00	0.16	0.13	0.09	0.06	0.03
6	0.00	1.00	0.09	0.08	0.06	0.05	0.03	0.02
7	1.00	0.00	0.00	0.00	0.00	0.00	0.00	0.00

3.9. Experimental Results for Fuzzification of Input/Output

1) The case of unimodal aggregated input MF

We use input value $X^{\prime }$ = 1080.

From (3.9), given (3.7) we have the following input MF

${\mu }_{X^{\prime }}$ (“1080”) = 1.00/0 + 0.97/1 + 0.93/2 + 0.89/3 + 0.84/4 + 0.78/5 + 0.71/6 + 0.59/7

Here is correspondent Figure 3 depiction.

Figure 3. MF of “1080” input value.

From (3.13) we are getting MFs for intersection of input MF of “1800” with the set of MFs of input centroid Table 2.

${\mu }_{X_0^{inter}}$ (“with 720”) = 1.00/0 + 0.88/1 + 0.75/2 + 0.63/3 + 0.50/4 + 0.38/5 + 0.25/6 + 0.13/7

${\mu }_{X_1^{inter}}$ (“with 1620”) = 0.97/0 + 0.97/1 + 0.93/2 + 0.89/3 + 0.84/4 + 0.78/5 + 0.71/6 + 0.59/7

${\mu }_{X_2^{inter}}$ (“with 3060”) = 0.79/0 + 0.97/1 + 0.79/2 + 0.60/3 + 0.44/4 + 0.30/5 + 0.18/6 + 0.09/7

${\mu }_{X_3^{inter}}$ (“with 5760”) = 0.39/0 + 0.56/1 + 0.77/2 + 0.89/3 + 0.77/4 + 0.56/5 + 0.39/6 + 0.25/7

${\mu }_{X_4^{inter}}$ (“with 8100”) = 0.29/0 + 0.42/1 + 0.56/2 + 0.70/3 + 0.84/4 + 0.78/5 + 0.71/6 + 0.59/7

${\mu }_{X_5^{inter}}$ (“with 10080”) = 0.13/0 + 0.23/1 + 0.35/2 + 0.49/3 + 0.65/4 + 0.78/5 + 0.71/6 + 0.59/7

${\mu }_{X_6^{inter}}$ (“with 11700”) = 0.21/0 + 0.35/1 + 0.48/2 + 0.59/3 + 0.70/4 + 0.78/5 + 0.71/6 + 0.59/7

${\mu }_{X_7^{inter}}$ (“with 12600”) = 0.35/0 + 0.50/1 + 0.61/2 + 0.71/3 + 0.79/4 + 0.78/5 + 0.71/6 + 0.59/7

We depicted results in Figure 4.

Figure 4. Intersection of “1080” input value MF with MFs of centroids.

From (3.16), given (3.15) we are getting the following fuzzy set ${\tilde{X}}^{aggr}$

${\mu }_{X^{aggr}}$ (“1080”) = 1.00/0 + 0.97/1 + 0.93/2 + 0.89/3 + 0.84/4 + 0.78/5 + 0.71/6 + 0.59/7

Here is correspondent Figure 5 depiction.

Linguistic classification of aggregated fuzzy set, represented by ${\mu }_{X^{aggr}}$ (“1080”).

2) The Case of Polymodal Aggregated Input MF

We use input value $X^{\prime }$ = 2430.

By using (3.8), given (3.7) we have the following input MF

${\mu }_{X^{\prime }}$ (“2430”) = 0.93/0 + 0.97/1 + 1.00/2 + 0.97/3 + 0.93/4 + 0.89/5 + 0.84/6 + 0.78/7

Here is correspondent Figure 6 depiction.

Figure 5. Aggregate input MF for “1080”.

Figure 6. MF of “2430” input value.

Again from (3.11) we are getting MFs for intersection of input MF of “2430” with the set of MFs of input centroids from Table 2.

${\mu }_{X_0^{inter}}$ (“with 720”) = 0.93/0 + 0.88/1 + 0.75/2 + 0.63/3 + 0.50/4 + 0.38/5 + 0.25/6 + 0.13/7

${\mu }_{X_1^{inter}}$ (“with 1620”) = 0.93/0 + 0.97/1 + 0.97/2 + 0.93/3 + 0.89/4 + 0.84/5 + 0.78/6 + 0.71/7

${\mu }_{X_2^{inter}}$ (“with 3060”) = 0.79/0 + 0.97/1 + 0.79/2 + 0.60/3 + 0.44/4 + 0.30/5 + 0.18/6 + 0.09/7

${\mu }_{X_3^{inter}}$ (“with 5760”) = 0.39/0 + 0.56/1 + 0.77/2 + 0.97/3 + 0.77/4 + 0.56/5 + 0.39/6 + 0.25/7

${\mu }_{X_4^{inter}}$ (“with 8100”) = 0.29/0 + 0.42/1 + 0.56/2 + 0.70/3 + 0.85/4 + 0.89/5 + 0.84/6 + 0.70/7

${\mu }_{X_5^{inter}}$ (“with 10080”) = 0.13/0 + 0.23/1 + 0.35/2 + 0.49/3 + 0.65/4 + 0.82/5 + 0.84/6 + 0.78/7

${\mu }_{X_6^{inter}}$ (“with 11700”) = 0.21/0 + 0.35/1 + 0.48/2 + 0.59/3 + 0.70/4 + 0.81/5 + 0.84/6 + 0.78/7

${\mu }_{X_7^{inter}}$ (“with 12600”) = 0.35/0 + 0.50/1 + 0.61/2 + 0.71/3 + 0.79/4 + 0.87/5 + 0.84/6 + 0.78/7

Here is correspondent Figure 7 depiction.

Figure 7. Intersection of “2430” Input Value MF with MFs of centroids.

One more time from (3.13), given (3.14) we are getting MF for the following fuzzy set ${\tilde{X}}^{aggr}$

${\mu }_{X^{aggr}}$ (“2430”) = 0.93/0 + 0.97/1 + 0.97/2 + 0.97/3 + 0.89/4 + 0.89/5 + 0.84/6 + 0.78/7

Here is correspondent Figure 8 depiction.

Figure 8. Aggregate input MF for “2430”.

Linguistic classification of aggregated fuzzy set, represented by ${\mu }_{X^{aggr}}$ (“2430”) from (3.28) means: $X^{aggr}$ = “in1” OR “in2” OR “in3”.

3.10. Use of Fuzzy Inference

1) The case of unimodal aggregated input MF

We use input value $X^{\prime }$ = 1080.

From (3.9), given (3.7) we have the following input MF

${\mu }_{X^{\prime }}$ (“1080”) = 1.00/0 + 0.97/1 + 0.93/2 + 0.89/3 + 0.84/4 + 0.78/5 + 0.71/6 + 0.59/7

We use binary relationship matrix from Table 3 in FCR (3.22) and getting the following fuzzy set ${\tilde{Y}}^{\prime }$ .

${\mu }_{Y^{\prime }}$ = 0.59/0 + 0.71/1 + 0.78/2 + 0.84/3 + 0.89/4 + 0.93/5 + 0.97/6 + 1.00/7

By use of the integration (3.24) and (3.25) we are getting

${\mu }_{Y_0^{inter}}$ (“with 10”) = 0.78/0 + 0.79/1 + 0.60/2 + 0.44/3 + 0.30/4 + 0.18/5 + 0.09/6 + 0.03/7

${\mu }_{Y_1^{inter}}$ (“with 28”) = 0.70/0 + 0.84/1 + 0.89/2 + 0.85/3 + 0.70/4 + 0.56/5 + 0.42/6 + 0.29/7

${\mu }_{Y_2^{inter}}$ (“with 48”) = 0.39/0 + 0.56/1 + 0.77/2 + 0.89/3 + 0.77/4 + 0.56/5 + 0.39/6 + 0.25/7

${\mu }_{Y_3^{inter}}$ (“with 61”) = 0.78/0 + 0.84/1 + 0.89/2 + 0.89/3 + 0.97/4 + 0.97/5 + 0.93/6 + 0.89/7

${\mu }_{Y_4^{inter}{}_{ }}$ (“with 66”) = 0.71/0 + 0.79/1 + 0.87/2 + 0.89/3 + 0.97/4 + 0.94/5 + 0.87/6 + 0.79/7

${\mu }_{Y_5^{inter}}$ (“with 78”) = 0.23/0 + 0.35/1 + 0.49/2 + 0.65/3 + 0.82/4 + 0.97/5 + 0.82/6 + 0.65/7

${\mu }_{Y_6^{inter}}$ (“with 93”) = 0.25/0 + 0.38/1 + 0.50/2 + 0.63/3 + 0.75/4 + 0.88/5 + 0.97/6 + 0.88/7

${\mu }_{Y_7^{inter}}$ (“with 103”) = 0.21/0 + 0.35/1 + 0.48/2 + 0.59/3 + 0.70/4 + 0.81/5 + 0.90/6 + 0.93/7

By use of the aggregation (3.26) and (3.27) we are getting in terms of singletons ${\mu }_{Y^{aggr}}\left(v_j\right)/v_j|\forall j\in \left[0,CardV\right]$ . See Figure 9.

${\mu }_{Y^{aggr}}$ = 0.59/0 + 0.71/1 + 0.78/2 + 0.84/3 + 0.89/4 + 0.93/5 + 0.97/6 + 1.00/7

Thus, from (3.28) linguistical-semantically it means: $Y^{aggr}$ = “out7”.

Here is how fuzzy conditional inference (3.18) transformed due to use of fuzzy clustering mechanism.

Ant 1: If $x$ is “in0”, then $y$ is “out7”

Ant2: $x^{\prime }$ is “in0”

--------------------------------------------

Cons: $y^{aggr}$ is “out7”

We use defuzzification procedure, given, that $\exists V_1\subseteq V=\left\{7\right\},Card{V}_1=1$ and from Table 2 $oc{l}_7=103$ .

Thus, from (3.30) we are getting $y_{Def}=103$ .

Figure 9. Aggregate output MF for “103”.

2) The Case of Polymodal Aggregated Input MF

We use input value $X^{\prime }$ = 2430.

By using (3.8), given (3.7) we have the following input MF.

One more time from (3.13), given (3.12) we are getting the following fuzzy set ${\tilde{X}}^{aggr}$

${\mu }_{X^{aggr}}$ (“2430”) = 0.93/0 + 0.97/1 + 0.97/2 + 0.97/3 + 0.89/4 + 0.89/5 + 0.84/6 + 0.78/7

from (3.28) linguistical-semantically it means: $X^{aggr}$ = “in1” OR “in2” OR “in3”.

Again, we use binary relationship matrix from Table 3 in FCR (3.22) and getting the following fuzzy set $\widetilde{Y^{\prime }}$

${\mu }_{{Y^{\prime }}_{ }}\left(v\right)$ = 0.78/0 + 0.84/1 + 0.89/2 + 0.89/3 + 0.97/4 + 0.97/5 + 0.97/6 + 0.93/7

One more time by use of the integration (3.24) and (3.25) we are getting

${\mu }_{Y_0^{inter}}$ (“with 10”) = 0.78/0 + 0.79/1 + 0.60/2 + 0.44/3 + 0.30/4 + 0.18/5 + 0.09/6 + 0.03/7

${\mu }_{Y_1^{inter}}$ (“with 28”) = 0.70/0 + 0.84/1 + 0.89/2 + 0.85/3 + 0.70/4 + 0.56/5 + 0.42/6 + 0.29/7

${\mu }_{Y_2^{inter}}$ (“with 48”) = 0.39/0 + 0.56/1 + 0.77/2 + 0.89/3 + 0.77/4 + 0.56/5 + 0.39/6 + 0.25/7

${\mu }_{Y_3^{inter}}$ (“with 61”) = 0.78/0 + 0.84/1 + 0.89/2 + 0.89/3 + 0.97/4 + 0.97/5 + 0.93/6 + 0.89/7

${\mu }_{Y_4^{inter}}$ (“with 66”) = 0.71/0 + 0.79/1 + 0.87/2 + 0.89/3 + 0.97/4 + 0.94/5 + 0.87/6 + 0.79/7

${\mu }_{Y_5^{inter}}$ (“with 78”) = 0.23/0 + 0.35/1 + 0.49/2 + 0.65/3 + 0.82/4 + 0.97/5 + 0.82/6 + 0.65/7

${\mu }_{Y_6^{inter}}$ (“with 93”) = 0.25/0 + 0.38/1 + 0.50/2 + 0.63/3 + 0.75/4 + 0.88/5 + 0.97/6 + 0.88/7

${\mu }_{Y_7^{inter}}$ (“with 103”) = 0.21/0 + 0.35/1 + 0.48/2 + 0.59/3 + 0.70/4 + 0.81/5 + 0.90/6 + 0.93/7

By use of the aggregation (3.26) and (3.27) we are getting in terms of singletons ${\mu }_{Y^{aggr}}\left(v_j\right)/v_j|\forall j\in \left[0,CardV\right]$ . See Figure 10.

${\mu }_{Y^{aggr}}$ () = 0.78/0 + 0.84/1 + 0.89/2 + 0.89/3 + 0.97/4 + 0.97/5 + 0.97/6 + 0.93/7

Thus, from (3.28) linguistical-semantically it means: $Y^{aggr}$ = “out4” OR “out5” OR “out6”.

Here is how fuzzy conditional inference (3.18) transformed due to use of fuzzy

clustering mechanism.

Ant 1: If $x$ is “in0”, then $y$ is “out7”

Ant2: $x^{\prime }$ is “in2”

--------------------------------------------

Cons: $y_{ }^{aggr}$ is “out4” OR “out5” OR “out6”

We use defuzzification procedure, given, that $\exists V_1\subseteq V=\left\{4,5,6\right\},Card{V}_1=3$ and from Table 2:

$oc{l}_4=66$ , $oc{l}_5=78$ and $oc{l}_6=93$ .

Thus, from (3.30) we are getting $y_{Def}=79$ .

Figure 10. Aggregate output MF for “79”.

4. Conclusion

In this study we examined EA non-linear scaled based approach to map original physical scale into a linguistical one. This approach allows to assign linguistic labels not to an evenly divided physical scale, but rather to arbitrary intervals within it, associated with expert’s knowledge and considerations about the nature of represented physical component. Thus, considered physical scale would be divided by certain number of linguistically labeled clusters. This approach makes fuzzy decision-making mechanism more properly applied to a nature of real-world behavior. In this study we covered issues of scaling, building fuzzy clusters, linguistic labeling, the fuzzification of a model’s input/output, the aggregation of fuzzy elements of a model and appropriate fuzzy decision-making mechanism. We also presented corresponding defuzzification mechanism.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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