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.

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)

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.

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.

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.

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

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

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)

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 ${\mu }_{X^{aggr}}\left(u_j\right)/u_j|\forall j\in \left[0,CardU\right]$

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)

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)

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 $\tilde{Y}$.

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)

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 .

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.

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 .

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.

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.

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.

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

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 $y_{Def}=103$.

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 $y_{Def}=79$.