Handling uncertainty in rheological properties of green eggshell nanocomposites by a fuzzy-hybrid modeling approach: a comparative study

White eggshells were obtained from a farm in Istanbul (Turkey). κ-carrageenan was (788,7 g mol⁻¹) and cetyltrimethylammonium bromide (CTAB) (Molecular Weight:364.45 g mol⁻¹) purchased from Sigma Aldrich Company (Germany). Polyvinyl alcohol (60.000  g mol⁻¹), acetic acid (purity: 100%), and ethanol (purity: 96%) were purchased from Merck Company (Germany).

The chemical and surface properties of the prepared ES NCs were determined using various characterization techniques such as scanning electron microscopy (SEM, FEI Company, USA) with a gold coating process at 20 kV of the accelerating voltage, dynamic light scattering (DLS-Mastersizer source material samples up to 3X trituration step), and fourier transform infrared spectroscopy (FTIR) in the frequency range from 400 to 4000 cm⁻¹. The surface tensions of ES NCs were performed using a force tensiometer (KRUSS Company, Germany). The z-average size of the ES NCs is computed by the following equation [20].

$$\begin{eqnarray}&&{D}_{{\rm{avg}}}=\displaystyle \frac{\displaystyle \sum {I}_{i}}{\displaystyle \sum \displaystyle \frac{{I}_{i}}{{D}_{i}}}\end{eqnarray} \tag{ 2.1 }$$

here, I is the intensity, D is the hydrodynamic diameter, and D_avg is the z-average size of the ES NCs.

The ES NCs were prepared following the experimental procedure described in our previous study [21]. 10 g of white eggshell was washed with distilled water and dried in a microwave oven. (1/1, v/v) 50 ml of water/ethanol solution was added to the eggshell and placed in the microwave oven for 5 min. Then, the solution was mixed with 10 ml of glacial acetic acid/distilled water (1:49, v/v) solution and sonicated at 35% amplitude for 1 h. 5.0 g of PVA were dissolved in 50 ml distilled water at 80 °C and kept for 5 days in a dark glass. κ-carrageenan (0.1 g) was dissolved in 50 ml distilled water. PVA, κ-carrageenan, and eggshell solutions were mixed and sonicated for 1 h. Finally, the sample was filtered through a sterile filter (0.22 micron) and stored in a sterile container at 25 °C. Additionally, the optimum pH, eggshell mass fractions (2%, 5%, and 10%), temperature (25 °C–50 °C), pH (3–10), CTAB mass fraction (2%, 4%, and 5%), variety, and concentration of the ES NCs were examined.

Regression analysis is an effective and reliable statistical method to determine the impact of one or several independent variable(s) on a dependent variable. This most widely used method has broad application areas such as and engineering, biology, business, and economic forecasting.

There are some cases that the observations are not exact and stable in nature. To deal with this vague and uncertain environment, fuzziness is considered in systems where human estimation is effective, and thus, a fuzzy structure of regression is developed. Fuzzy linear regression analysis is first introduced by Tanaka et al [22]. They formulated a linear programming problem to determine the regression coefficients in fuzzy form. After then, several articles have addressed the issue of regression in a fuzzy environment in the literature [18, 23].

In this paper, we used the fuzzy regression, which minimizes the total fuzziness of the model proposed by Lee and Tanaka (1999) [24], referred to as possibilistic linear regression. They used a fuzzy linear function, of which only parameters are fuzzy numbers, as a regression model to express fuzziness. They proposed a fuzzy regression model by assuming nonsymmetric triangular fuzzy numbers. In this regression model, the dataset contains crisp inputs and after the application of the model, fuzzy outputs are obtained. In the proposed fuzzy regression analysis, quadratic programming problem is used, which integrates the properties of central tendency and possibility. The proposed model minimizes the total spread, whereas its constraints show that the observed values should be covered by the estimated fuzzy number at the particular level of threshold. The quadratic programming approach allows to obtain the center and the spread coefficients simultaneously and presents some trade-offs between the minimum spread and the central tendency in the regression model.

Consider $\left({x}_{j};{y}_{j}\right)=\left(1,{x}_{j1},\,\ldots ,\,{x}_{jn};{y}_{j}\right),j=1,\,\ldots ,\,m$ is the observed dataset where ${x}_{j}=\left(1,{x}_{j1},\,\ldots ,\,{x}_{jn}\right)$ is the ${j}^{th}$ input data, ${y}_{j}$ is the ${j}^{th}$ output data and $m$ is the data size. The fuzzy regression model is presented as

$$\begin{eqnarray}&&\tilde{Y}\left({\boldsymbol{x}}\right)={\tilde{A}}_{0}+{\tilde{A}}_{1}{x}_{1}+\cdots +{\tilde{A}}_{n}{x}_{n}\end{eqnarray} \tag{ 2.2 }$$

where the components of input data ${\boldsymbol{x}}=\left(1,{x}_{1},\,\ldots ,\,{x}_{n}\right)$ are crisp numbers, the coefficients ${\tilde{A}}_{i},i=0,\,\ldots ,\,n$ and the estimated output $\tilde{Y}\left({\boldsymbol{x}}\right)$ are (non-symmetric) triangular fuzzy numbers. A triangular fuzzy number ${\tilde{A}}_{i}=\left({a}_{i},\,{l}_{i},\,{r}_{i}\right)$ is defined by using a membership function ${\mu }_{{\tilde{A}}_{i}}\left(x\right)$ which takes a value between 0 and 1, such that

$$\begin{eqnarray}{\mu }_{{\tilde{A}}_{i}}\left(x\right)=\left\{\begin{array}{c}\begin{array}{cc}1-\displaystyle \frac{{a}_{i}-x}{{l}_{i}}, & if{a}_{i}-{l}_{i}\leqslant x\leqslant {a}_{i}\end{array}\\ \begin{array}{cc}1-\displaystyle \frac{x-{a}_{i}}{{r}_{i}}, & if{a}_{i}\leqslant x\leqslant {a}_{i}+{r}_{i}\end{array}\\ \begin{array}{cc}0, & otherwise\end{array}\end{array}\right.\end{eqnarray} \tag{ 2.3 }$$

where ${a}_{i}$ is the center, ${l}_{i}$ and ${r}_{i}$ are left and right spreads, respectively. Therefore, the regression model given in (1) can be expressed as

$$\begin{eqnarray}\begin{array}{rcl}\tilde{Y}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right) & = & \left({a}_{0},\,{l}_{0},\,{r}_{0}\right)+\left({a}_{1},\,{l}_{1},\,{r}_{1}\right){x}_{j1}+\cdots +\left({a}_{n},\,{l}_{n},\,{r}_{n}\right){x}_{jn}\\ & = & \left({\theta }_{C}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right),{\theta }_{L}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right),{\theta }_{R}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)\right)\end{array}\end{eqnarray} \tag{ 2.4 }$$

where,

$$\begin{eqnarray}&&\begin{array}{c}{\theta }_{C}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)=\displaystyle \sum _{i=0}^{n}{a}_{i}{x}_{ji}\\ {\theta }_{L}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)=\displaystyle \displaystyle \sum _{{x}_{ji}\geqslant 0}{l}_{i}{x}_{ji}-\displaystyle \displaystyle \sum _{{x}_{ji}\lt 0}{r}_{i}{x}_{ji}\\ {\theta }_{R}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)=\displaystyle \displaystyle \sum _{{x}_{ji}\geqslant 0}{r}_{i}{x}_{ji}\,-\,\displaystyle \displaystyle \sum _{{x}_{ji}\lt 0}{l}_{i}{x}_{ji}\end{array}\end{eqnarray} \tag{ 2.5 }$$

represent the center, the left-spread, and the right-spread of the fuzzy output $\tilde{Y}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right),$ respectively. It should be noted that the left and right spreads of the fuzzy output, i.e. ${\theta }_{L}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)\,$and ${\theta }_{R}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right),$ are positive since the left and right spreads of the fuzzy coefficients, i.e. ${l}_{i}$ and ${r}_{i},$ $i=0,\,\ldots ,\,n,$ are assumed to be positive.

Since the parameter $h$ is a given threshold, the $h-\,$level set of the estimated output $\tilde{Y}\left({\boldsymbol{x}}\right)$ is determined as an interval such that

$$\begin{eqnarray}&&{\left[\tilde{Y}\left({\boldsymbol{x}}\right)\right]}_{h}=\left[{\theta }_{C}\left({\boldsymbol{x}}\right)-\left(1-h\right){\theta }_{L}\left({\boldsymbol{x}}\right),{\theta }_{C}\left({\boldsymbol{x}}\right)+\left(1-h\right){\theta }_{R}\left({\boldsymbol{x}}\right)\right]\end{eqnarray} \tag{ 2.6 }$$

To determine the optimal fuzzy coefficients ${\tilde{A}}_{i},i=0,\,\ldots ,\,n$ of the fuzzy regression model, the sum of spreads of the estimated fuzzy outputs $\tilde{Y}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right),j=1,\,\ldots ,\,m$ is considered as

$$\begin{eqnarray}&&\left(1-h\right)\displaystyle \sum _{j=1}^{m}\left({\theta }_{L}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)+{\theta }_{R}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)\right)\end{eqnarray} \tag{ 2.7 }$$

and the sum of squared distances between the estimated output centers ${\theta }_{C}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)$ and the observed outputs, which is known as the least squares method, is presented as

$$\begin{eqnarray}&&\displaystyle \sum _{j=1}^{m}{\left({\theta }_{C}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)-{\boldsymbol{a}}{{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)}^{2}\end{eqnarray} \tag{ 2.8 }$$

where ${\boldsymbol{a}}$ is a center vector.

The fuzzy regression model aims to minimize the sum of equations (2.6) and (2.7), which reflect both properties of fuzzy regression and least squares, subject to including all the given data and a threshold $h.$ Therefore, the mathematical model of the fuzzy regression is presented as

$$\begin{eqnarray*}&&\min \,{w}_{1}\displaystyle \sum _{j=1}^{m}{\left({\theta }_{C}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)-{\boldsymbol{a}}{{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)}^{2}+{w}_{2}\left(1-h\right)\displaystyle \sum _{j=1}^{m}\left({\theta }_{L}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)+{\theta }_{R}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)\right)\end{eqnarray*}$$

s.t.

$$\begin{eqnarray*}&&{\theta }_{C}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)+\left(1-h\right){\theta }_{R}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)\geqslant {y}_{j}\end{eqnarray*}$$

${\theta }_{C}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)-\left(1-h\right){\theta }_{L}\left({{\boldsymbol{x}}}_{{\boldsymbol{j}}}\right)\leqslant {y}_{j},\,j=1,\,\ldots ,\,m$

$$\begin{eqnarray*}&&{l}_{i}\geqslant 0,\,{r}_{i}\geqslant 0,\,i=1,\,\ldots ,\,n\end{eqnarray*}$$

where ${w}_{1}$ and ${w}_{2}$ are weight coefficients.

In this study, R programming language is used with R-Studio version 1.4.1106 to obtain the fuzzy regression results. We have utilized the PLRLS method of the fuzzyreg software package [24, 25].

2.5.1. Data

2.5.2. Determination of critical micelle concentration

2.5.3. The proposed crisp model

2.5.4. The proposed fuzzy-hybrid model

Primarily, the R² [30] values are evaluated for each regression. Then, correlation diagrams are presented for all constructed mathematical models, where experimental values are plotted against the calculated ones. Then, in order to provide quantitative statistical error analysis, the sum of squares of errors (SSE) [17], the hybrid fractional error function (HYBRID) [31], Marquart's percentage standard deviation (MPSD) [32], and the average relative error (ARE) [33], are evaluated for the crisp models and the central tendency of the fuzzy models. The mathematical formulations of R², SSE, HYBRID, MPSD, and ARE are given in equations (2.9–2.13), respectively.

$$\begin{eqnarray}&&{R}^{2}=1-\displaystyle \frac{\left({\sum }_{i=1}^{n}{\left({\gamma }_{i}-{\gamma }_{{\rm{cal}}}\right)}^{2}\right)}{\left({\sum }_{i=1}^{n}{\left({\gamma }_{i}-{\gamma }_{{\rm{avg}}}\right)}^{2}\right)}\end{eqnarray} \tag{ 2.9 }$$

$$\begin{eqnarray}&&SSE=\displaystyle \sum _{i=1}^{n}{\left({\gamma }_{cal}-{\gamma }_{e}\right)}_{i}^{2}\end{eqnarray} \tag{ 2.10 }$$

$$\begin{eqnarray}&&HYBRID=\displaystyle \frac{100}{n-p}\displaystyle \sum _{i=1}^{n}\left[\displaystyle \frac{{\left({\gamma }_{e}-{\gamma }_{cal}\right)}_{i}^{2}}{{\gamma }_{e}}\right]\end{eqnarray} \tag{ 2.11 }$$

$$\begin{eqnarray}&&MPSD=100\sqrt{\displaystyle \frac{1}{n-p}\displaystyle \sum _{i=1}^{n}{\left(\displaystyle \frac{{\gamma }_{e}-{\gamma }_{cal}}{{\gamma }_{e}}\right)}_{i}^{2}}\end{eqnarray} \tag{ 2.12 }$$

$$\begin{eqnarray}&&ARE=\displaystyle \frac{100}{n-p}\displaystyle \sum _{i=1}^{n}{\left|\displaystyle \frac{{\gamma }_{e}-{\gamma }_{cal}}{{\gamma }_{e}}\right|}_{i}\end{eqnarray} \tag{ 2.13 }$$

here, ${\gamma }_{{\rm{e}}}:$ experimental surface tension, ${\gamma }_{{\rm{cal}}}:$ calculated surface tension, ${\gamma }_{{\rm{avg}}}:$ mean of experimental surface tension, n: number of data points and p: number of parameters.

Recently, researchers have investigated in detail the rheological, surface, and chemical properties of advanced nanostructures to increase biocompatibility and biodegradability [34–37]. To investigate the surface and chemical properties of ES NCs, different techniques such as SEM, FTIR, and DSC were used. In figures 1(a)–(c), the SEM, DLS, and FTIR results of ES NCs were presented. The SEM image of the prepared ES NCs was evaluated to determine the distribution, morphology, and particle size of ES NCs. According to the SEM result, it was clear that the ES NCs had a had a porous flower-shaped nanostructure which has a shape of a coral reef structure. The structure of ES NCs exhibited a structure of coral reefs with micro/mesopores in between the nanolayers and three-dimensional (3D) hierarchical porous structure. It was assumed that the growth of EC had aggregation mechanisms driving the self-assembly of 3D structure (https://doi.org/10.1016/j.est.2021.102749, Journal of Energy Storage Volume 40, August 2021, 102749) Additionally, the ES NCs had a particle distribution with diameters of about 100–200 nm (figure 1(a)). The DLS technique was used to measure the hydrodynamic diameter of the ES NCs in solution. According to the DLS results, the z-average size of ES NCs was 101.97 nm with a homogenous particle distribution, having a uniform layered nano-porous structure approved by the SEM and DLS results. Furthermore, it was found that the ES NCs had a large surface area of 220.2 m² g⁻¹.

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In our previous study, the FTIR technique was used to identify the chemical structural and functional groups of ES NCs [21]. The FTIR result was presented in figure 1(c). We observed at 3400 cm⁻¹−3199 cm⁻¹ (–OH), 2913 cm⁻¹− 2856 cm⁻¹ (–CH stretching), 1726 cm⁻¹−1640 cm⁻¹ (–C=O), 1455 cm⁻¹−1412 cm⁻¹, 1369 cm⁻¹, 1100 cm⁻¹ and 1050 cm⁻¹ (–C=O, carbonate) [21].

According to the FTIR spectra, it was concluded that various surface functional groups (–C=O, and O-H groups) on the surface of the nanostructre.

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In this study, rheological properties of nanostructures were investigated in detail at different operating parameters such as the mass fraction of eggshell, the mass fraction of surfactant, pH, and temperature and were shown in figures 3, 4. The surface tension results of ES NCs/water solutions were calculated between 39.22 mN m⁻¹ to 63.11 mN m⁻¹ in a range of 4–50 ppm at 25 °C. According to rheological results, we found that the minimum surface tension value was calculated for the ES NCs (2% eggshell) and showed that there was a correlation between the mass fraction of the nanostructure and surface tension (see figures 5, 6). However, the lowest surface tension was found to be 39.22 mN m⁻¹ for the ES NCs (5% CTAB). Consequently, the surface tension values were changed with temperature, and we showed that the optimum temperature was 25 °C and pH 5 [38, 39].

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Critical micelle concentrations (CMC) are determined by the intersection of the lines which are obtained by linear regressions relative to the data, all illustrated in figure 2. The CMCs are determined as 33.92 ppm, 26.77 ppm, and 22.42 ppm for mass fractions of 2%, 5%, and 10% eggshell, respectively.

In this study, we have obtained surface tension as a function of operating variables concentration (C), temperature (T), pH, and CTAB mass fraction (m_CTAB, %). Firstly, an exponential regression is applied to the surface tension data corresponding to concentration, while the remaining variables are kept constant at T = 25 °C and pH = 5. The functions of ${\gamma }_{C2}(C)$ equation (3.1), ${\gamma }_{C5}(C)\,$equation (3.2), and ${\gamma }_{C10}(C)$ equation (3.3) are provided for mass fraction of 2%, 5%, and 10% eggshell, respectively. These functions are illustrated with the experimental data in figure 3. The R² value for these equations are 0.96, 0.97, and 0.99, respectively.

$$\begin{eqnarray}&&{\gamma }_{C2}\left(C\right)=33.23014+3.01112\,{e}^{\displaystyle \frac{C}{21.07341}}\end{eqnarray} \tag{ 3.1 }$$

$$\begin{eqnarray}&&{\gamma }_{c5}\left(C\right)=42.76296+0.70349\,{e}^{\displaystyle \frac{C}{14.76296}}\end{eqnarray} \tag{ 3.2 }$$

$$\begin{eqnarray}&&{\gamma }_{c10}\left(C\right)=48.77553+0.0432\,{e}^{\displaystyle \frac{C}{8.09506}}\end{eqnarray} \tag{ 3.3 }$$

To determine surface tension as a function of temperature, pH, and m_CTAB, x linear regressions are applied to the corresponding data as illustrated in figure 4. The relationships between surface tension—temperature (R² = 0.97), surface tension—pH (R² = 0.96), and surface tension—m_CTAB(R² = 0.94), are presented in equations (3.4), (3.5), and equation (3.6), respectively.

$$\begin{eqnarray}&&{\gamma }_{T}\left(T\right)=-0.35156\,T+59.25214\end{eqnarray} \tag{ 3.4 }$$

where pH = 5, and the mass fraction is 5% (m_CTAB = 5).

$$\begin{eqnarray}&&{\gamma }_{pH}\left(pH\right)=1.36564\,pH+42.91471\end{eqnarray} \tag{ 3.5 }$$

where T = 25 °C, and the mass fraction is 5% (m_CTAB = 5).-

$$\begin{eqnarray}&&{\gamma }_{{m}_{CTAB}}\left({m}_{CTAB}\right)=-6.39786\,{m}_{CTAB}+73.10214\end{eqnarray} \tag{ 3.6 }$$

where T = 25 °C, and pH = 5.

After the obtainment of equations (3.4–3.6), the functions ${\gamma }_{T}\left(T\right),$ ${\gamma }_{pH}\left(pH\right)\,$are normalized as in equations (3.7–3.9), respectively. The normalization is performed so that the normalized functions result 1 at fixed parameters T = 25 °C, pH = 5, and m_CTAB = 5.

$$\begin{eqnarray}&&{\gamma }_{TN}\left(T\right)=\displaystyle \frac{{\gamma }_{T}\left(T\right)}{{\gamma }_{T}\left(25\right)}=-6.9667\cdot {10}^{-3}T+1.17417\end{eqnarray} \tag{ 3.7 }$$

$$\begin{eqnarray}&&{\gamma }_{pHN}\left(pH\right)=\displaystyle \frac{{\gamma }_{pH}\left(pH\right)}{{\gamma }_{pH}\left(5\right)}=2.74540\cdot {10}^{-2}pH+0.86273\end{eqnarray} \tag{ 3.8 }$$

$$\begin{eqnarray}&&{\gamma }_{mN}\left({m}_{CTAB}\right)=-\displaystyle \frac{{\gamma }_{m}\left({m}_{CTAB}\right)}{{\gamma }_{m}\left(5\right)}=-0.15562\,{m}_{CTAB}+1.77807\,\end{eqnarray} \tag{ 3.9 }$$

Finally, the multiplication of equations (3.7–3.9) with the functions ${\gamma }_{C2}\left(C\right),$ ${\gamma }_{C5}\left(C\right),$ and ${\gamma }_{C10}\left(C\right)$ (equations (3.1–3.3)) yields the surface tension as a function of the operating variables for mass fractions of 2%, 5% and 10% eggshell, respectively equations (3.10–3.12).

$$\begin{eqnarray}&&{\gamma }_{2 \% }\left(T,pH,{m}_{CTAB},C\right)={\gamma }_{TN}\left(T\right){\gamma }_{pHN}\left(pH\right){\gamma }_{mN}\left({m}_{CTAB}\right){\gamma }_{C2}\left(C\right)\end{eqnarray} \tag{ 3.10 }$$

$$\begin{eqnarray}&&{\gamma }_{5 \% }\left(T,pH,{m}_{CTAB},C\right)={\gamma }_{TN}\left(T\right){\gamma }_{pHN}\left(pH\right){\gamma }_{mN}\left({m}_{CTAB}\right){\gamma }_{C5}(C)\end{eqnarray} \tag{ 3.11 }$$

$$\begin{eqnarray}&&{\gamma }_{10 \% }\left(T,pH,{m}_{CTAB},C\right)={\gamma }_{TN}\left(T\right){\gamma }_{pHN}\left(pH\right){\gamma }_{mN}\left({m}_{CTAB}\right){\gamma }_{C10}\left(C\right)\end{eqnarray} \tag{ 3.12 }$$

While the obtained equations (3.10–3.12) may be utilized as a sufficient model for surface tension, there is possible improvement on the model by considering the fuzzy regression. Thus, to provide a more sophisticated mathematical model containing the random error in the data, a fuzzy-hybrid model is proposed in this study. Firstly, fuzzy linear regression is fitted to represent the surface tension data corresponding to concentration, as opposed to the exponential regression. For the values below the critical micelle concentration, the crisp linear regression represents the data adequately. Consequently, the fuzzy and crisp linear regressions are applied to the data above and below the critical micelle concentration, respectively. The fuzzy parameters obtained by the regression are represented in table 1. Note that the obtained M₂, A₅, and A₁₀ are crisp values since their spreads are zero.

By combining the fuzzy and linear regressions applied above and below C_cmc values, we obtain the piecewise functions in equations (3.13–3.15) for 2%, 5%, and 10% eggshell mass fraction, respectively. As the results of the proposed fuzzy-hybrid model, the following piecewise functions with the corresponding data are illustrated in figure 5. Since the same data is utilized in figures 3 and 5, it can be observed that the performance of the proposed fuzzy-hybrid model is superior to the proposed crisp model.

$$\begin{eqnarray}&&{\gamma }_{C2P}\left(C\right)=\left\{\begin{array}{c}39.1212+0.00862\,C,C\lt {C}_{CMC}=22.42\\ {A}_{2}+{M}_{2}C,C\geqslant {C}_{CMC}\end{array}\right.\end{eqnarray} \tag{ 3.13 }$$

$$\begin{eqnarray}&&{\gamma }_{C5P}\left(C\right)=\left\{\begin{array}{c}39.1212+0.00862\,C,C\lt {C}_{CMC}=26.77\\ {A}_{5}+{M}_{5}C,C\geqslant {C}_{CMC}\end{array}\right.\end{eqnarray} \tag{ 3.14 }$$

$$\begin{eqnarray}&&{\gamma }_{C10P}\left(C\right)=\left\{\begin{array}{c}39.1212+0.00862\,C,C\lt {C}_{CMC}=33.92\\ {A}_{10}+{M}_{10}C,C\geqslant {C}_{CMC}\end{array}\right.\end{eqnarray} \tag{ 3.15 }$$

The equivalent representation of the fuzzy relations, namely, central tendency, lower boundary, and upper boundary of the regression models are presented in table 2. The lower and upper boundaries provide a sensitivity approximation of the regression. Thus, the experimental error intervals can directly be determined by using these equations.

Table 2. The lower boundary, central tendency, and upper boundary equations of the respective fuzzy relations.

Fuzzy Relation	Equivalency Type	Equation
${A}_{2}+{M}_{2}\,C$	Lower Boundary	$15.1868+0.9474\,C$
	Central Tendency	$16.9010+0.9474\,C$
	Upper Boundary	$18.7933+0.9474\,C$
${A}_{5}+{M}_{5}C$	Lower Boundary	$22.5791+0.8162\,C$
	Central Tendency	$22.5791+0.8401\,C$
	Upper Boundary	$22.5791+0.8627\,C$
${A}_{10}+{M}_{10}\,C$	Lower Boundary	$12.7958+1.0348\,C$
	Central Tendency	$12.7958+1.0811\,C$
	Upper Boundary	$12.7958+1.1285\,C$

Finally, the constructed piecewise functions ${\gamma }_{C2P}\left(C\right),$ ${\gamma }_{C5P}\left(C\right),$ ${\gamma }_{C2P}\left(C\right)$ equations (3.13–3.15) may be multiplied by equations (3.7–3.9), to yield the fuzzy-hybrid model, resulting surface tension as a function of all the operating variables for mass fractions of 2%, 5%, and 10% eggshell, respectively equations (3.16–3.18).

$$\begin{eqnarray}&&{\gamma }_{2 \% F}\left(T,pH,{m}_{CTAB},C\right)={\gamma }_{TN}\left(T\right){\gamma }_{pHN}\left(pH\right){\gamma }_{mN}\left({m}_{CTAB}\right){\gamma }_{C2P}\left(C\right)\end{eqnarray} \tag{ 3.16 }$$

$$\begin{eqnarray}&&{\gamma }_{5 \% F}\left(T,pH,{m}_{CTAB},C\right)={\gamma }_{TN}\left(T\right){\gamma }_{pHN}\left(pH\right){\gamma }_{mN}\left({m}_{CTAB}\right){\gamma }_{C5P}(C)\end{eqnarray} \tag{ 3.17 }$$

$$\begin{eqnarray}&&{\gamma }_{10 \% F}\left(T,pH,{m}_{CTAB},C\right)={\gamma }_{TN}\left(T\right){\gamma }_{pHN}\left(pH\right){\gamma }_{mN}\left({m}_{CTAB}\right){\gamma }_{C10P}(C)\end{eqnarray} \tag{ 3.18 }$$

Statistical error analysis is conducted for functions obtained in equations (3.10–3.12) and (3.16–3.18) to evaluate the performance of the models obtained. SSE, HYBRID, MPSD and ARE values for each function are determined by comparing the results of each equation with 106 experimental data points. The experimental surface tension values plotted against the calculated ones are illustrated in figure 6. It shows that the errors are contained within the fuzzy coefficients. Indeed, y = x line is contained within the left and right spreads of the computed fuzzy number, i.e., fuzzy output.

In order to provide quantitative comparability, the error terms are computed only for the central tendency of the fuzzy model and given in table 3. It represents the error values in an acceptable range for both crisp and fuzzy models with comparison to our previous study [19]. For 2% and 5% eggshell mass fractions, the error values were found to be significantly lower, even if the fuzzy hybrid model was constructed by considering only the central tendency. For 10% eggshell mass fraction, even though the crisp model has given slightly better results, the comparison is with the central tendency of the fuzzy-hybrid model only. Since the output of the proposed fuzzy-hybrid model contains the errors as well, the fuzzy-hybrid model is highly preferable to the crisp model.