Mechanical characterization of particle reinforced jute fiber composite and development of hybrid Grey-ANFIS predictive model

ABSTRACT

Natural fiber composites are a potential material in a range of engineering applications because of their excellent properties, which include reduced weight, better strength, and economic affordability. Aside from these features, these materials are biodegradable and renewable. The intent of this study is to look through into mechanical properties of aluminum oxide (Al₂O₃), boron carbide (B₄C), and silicon carbide (SiC) particle-filled jute fiber polymer composite. The response surface methodology (RSM) with three levels/three factors is used to achieve the different combinations of process variables required to fabricate the desired polymer composites. In this regard, the effect of process variables on tensile characteristics, increase in weight %, and flexural characteristics is examined in detail. Further, the best combination of process parameters is chosen to produce composites with the desired mechanical qualities. The significance of such variables on each output variable is assessed using analysis of variance. A hybrid grey-based adaptive neuro-fuzzy inference system model is constructed for establishing multiple performance indexes. From the validation outcomes obtained, it is proved that the evolved model is proficient for precise prediction.

摘要

天然纤维复合材料由于其优异的性能，包括重量减轻、强度更好和经济实惠，是一种潜在的工程应用材料.除了这些特点，这些材料是可生物降解的和可再生的.本研究旨在深入研究氧化铝（Al2O3）、碳化硼（B4C）和碳化硅（SiC）颗粒填充黄麻纤维聚合物复合材料的力学性能.具有三个水平/三个因素的响应面方法（RSM）用于实现制造所需聚合物复合材料所需的工艺变量的不同组合.在这方面，详细检查了工艺变量对拉伸特性、重量增加（%）和弯曲特性的影响.此外，选择工艺参数的最佳组合以生产具有所需机械质量的复合材料.使用方差分析评估这些变量对每个输出变量的重要性.为了建立多个性能指标，构建了一种基于混合灰色的自适应神经模糊推理系统（ANFIS）模型.从所获得的验证结果来看，证明了进化模型能够准确预测.

KEYWORDS:

• Jute fiber
• particle
• RSM
• mechanical properties
• hybrid adaptive neuro-fuzzy inference system model
• ANOVA

关键词:

• 黄麻纤维
• 微粒
• 机械性能
• 混合自适应神经的模糊推理系统模型

Introduction

Natural fiber–reinforced composites (NFRCs) have attracted considerable attention from manufacturers and academics in recent decades for leveraging these materials’ applications for several industrial uses due to their lower cost, lightweight, and biodegradability. The exaction for and use of natural fiber is increasing due to environmental concerns. In this context, few researchers have explored the implications of life cycle and environmental impact evaluations on the processing of NFRCs in a literature study on the same. Researchers have found that NFRCs have superior mechanical properties such as flexural, tensile, impact, and interlaminar strengths. Further, they exhibit the best heat, water absorption, and tribological properties, making them a better choice for various structural applications (Sanjay et al. 2018; Girijappa et al. 2019; Ramesh et al. 2020; Vinod et al. 2020). Delamination may appear in natural fibers due to the lessened interference bonding amid the fiber and the polymer which affects the mechanical properties. Hence, there is a need to create hybrid laminated composites with the use of natural and synthetic fibers to enhance the required mechanical properties where the stacking order has been discovered to have a indicative effect on the desired characteristics (Balakrishnan et al. 2016; Feng et al. 2019; Karthi et al. 2019; Khan, Srivastava, and Gupta 2018; Sabeel Ahmed et al. 2012; Das et al. 2021). In specific, biofibers are viewed as an associable alternative to carbon fiber–reinforced composites in automobile applications. Jute fiber is an eminent and popular biofiber, considering its typical characteristics such as dimensional stability, ease of availability, and exceptional mechanical properties (Krishnasamy et al. 2020; Kumar, Nandal, and Gupta 2021; Lotfi et al. 2021). Jute fiber is a dominant material that significantly improves the quality of components in several applications, particularly in construction of buildings. Researchers have found that the fiber boosts tensile and compressive strengths (Maraveas 2020; Maslinda et al. 2017; Monteiro et al. 2016; Paul, Gouda, and Bhowmik 2021). Also, jute fabrics coated with bitumen emulsion and unsaturated polyester, which were exposed to γ-radiation, exhibited superior properties (Akter et al. 2018, 2020). So far, researchers investigated the properties of fabricated composites. However, few researchers studied the tribological behavior of a jute fiber composites according to the ASTM standards. The investigation revealed that fiber bonding significantly impacts tribological properties (Chaudhary, Kumar Bajpai, and Maheshwari (2018); (Raju et al. 2013; Singh and Zafar 2020; Yallew, Kumar, and Singh 2015). Jute fiber is a dominant material that significantly improves the properties in a variety of applications, particularly in building construction.

Consequently, the composites were reinforced to improve the desirable properties. However, fillers significantly minimize the downside of the reinforced fibers and further enhance the desired properties. (Jagadeesh et al. 2020; Akter et al. 2020)). Various in-organic fillers such as aluminum oxide, boron carbide, silicon carbide, and graphene were studied in this regard. Compared to other fillers, boron carbide exhibited the best elastic properties (Sayer 2014), and studies on wear phenomenon on inclusion of aluminum oxide portrayed a remarkable decrease in wear rate (Arpitha et al. 2017). Also, flexural and tensile strength was enhanced with significant proportions of fillers (Vinay et al. 2021; Hemath et al. 2020; Narayana and Gopalan 2022). Likewise, the inclusion of graphene powder exhibited a considerable increase in composite strength (Ganapathy et al. 2021). However, natural materials such as rosewood dust and padauk wood dust (Dinesh et al. 2020), carbonized coconut shell powder (Jenish et al; Akter et al. 2020), wood apple shell powder (Nagaraja Setty et al. 2020, 2021), wood charcoal (Das et al. 2021), fish bone, and egg shell powders as a filler materials were also endeavored (Abhishek et al. 2018; Ganesan et al. 2018). The properties of the prepared laminates were evaluated using the response surface methodology (RSM). With the use of statistical and mathematical approaches, a model was developed for analyzing a large number of input variables (Venkatachalam et al. 2021).

Determining the suitable process variable to achieve better performance in manufacturing is difficult for all industries. While investigating the multiple performance measures, the enhancement of one parameter may influence the other parameter. Multi-criteria decision-making (MCDM) is one of the best alternative approaches for determining the solutions in various applications. Grey relational grade (GRA) is one among the MCDM approach used in solving these kinds of problems (Kumar et al. 2021; Rajak et al. 2019; Rao et al. 2019). Even though GRA has various merits, the information obtained from this approach is unclear. To avoid such demerits, a decision-making tool supported by artificial intelligence was developed and implemented, resulting in a significant improvement in the process. ANFIS (adaptive neuro-fuzzy inference system) is a tool for accomplishing the desired objectives by amalgamating the benefits of neural network and fuzzy systems (Kharwar, Kumar Verma, and Singh 2020; Manikandan et al. 2019; Prakash et al. 2022; Thejasree and Krishnamachary 2022; Manikandan et al. 2019).

Developing and analyzing the process variables in the preparation of jute fiber composites need considerable attention. In this regard, variables (particle type, size, and percentage weight) as well as dependent variables (tensile stress and modulus, increase in weight %, flexural stress, and modulus) were given much attention in this work. Furthermore, an attempt is made to construct a decision-making tool (Hybrid Intelligent Decision-Making Tool-Grey ANFIS), which is being used to foresight the desired output measure.

Materials and methods

Jute woven fiber, LY-556 epoxy resin, and XY-54 hardener were procured from Vruksha Natural products, Guntur, while the ceramic particles were obtained from Nice Chemicals, Trichy, Tamil Nadu. The woven fiber is chopped according to the mold’s dimensions. Composites made of woven jute fiber and ceramic particles are produced using a mold box with dimensions of 300 × 300 × 3 mm³. These composite laminates are fabricated with the precise measurements needed. The entire process starts by introducing the resin into the dye. The mold is then sealed, and pressure is applied on the top side of the other plate which is used to enclose the bottom dye.

RSM is an inimitable technique used for modeling. RSM analyzes the data mathematically, statistically, and graphically, and it produces one or many outputs that are predisposed by different input variables as well. Also, the RSM approach works to develop mathematical relationships. The experiments in this work were created using the Box–Behnken Design (BBD) method. Accordingly, the design is obtained and the composites are fabricated. The various levels of process variables that are chosen are mentioned in Table 1. The Instron machine-make 8801 was used for the tensile and flexural testing, and the specimens were prepared in compliance with ASTM standards as D3039-15 and D790-15, respectively, as shown in Figures 1 and 2. The water absorption test is carried out in line with ASTM D 570 specifications.

Figure 1. Testing setup for tensile test.

Figure 2. Testing setup for flexural test.

Table 1. Process parameters.

Variables/ Levels	Level
	Low	Medium	High
	-1	0	1
Particle type	Sic	B4C	Al2O3
Particle size (in µm)	53	75	106
Particle weight percentage	4	8	12

Table 2 outlines the sequence of experiments that must occur in order for the samples to be prepared.

Table 2. Combination of process variables for composite preparation with coded/uncoded values.

Factors/ Exp. No.	Particle type		Particle size (µm)		Particle weight %
	Coded (A)	Actual	Coded (B)	Actual	Coded (C)	Actual
1	0	B4C	1	106	-1	4
2	1	Al2O3	−1	53	0	8
3	−1	Sic	0	75	−1	4
4	1	Al2O3	0	75	−1	4
5	−1	Sic	0	75	1	12
6	1	Al2O3	0	75	1	12
7	−1	Sic	−1	53	0	8
8	0	B4C	−1	53	−1	4
9	0	B4C	1	106	1	12
10	0	B4C	−1	53	1	12
11	−1	Sic	1	106	0	8
12	1	Al2O3	1	106	0	8
13	0	B4C	0	75	0	8
14	0	B4C	0	75	0	8
15	0	B4C	0	75	0	8

ANFIS model

Despite its numerous advantages, the output measure in the GRA method is better and commonly used for multi-aspect optimization, containing some unclear and uncertain information. With cutting-edge intelligent decision-making tools, the conventional MCDM approach may be supported. Combining these tools enhances process control, which leads to better performance. A strong model with the least amount of error was combined to create the ANFIS, a smart evaluation tool. Some soft computing strategies are more practical when precise mathematical statistics are not available. The soft computing approach deviates from the conventional one in that it yields accurate results and lowers ambiguity.

Results and discussions

The RSM-BBD methodology is used to prepare composites with different combinations. A total of 15 composite specimens were prepared, and the samples are mechanically tested to determine the desired mechanical characteristics. This section explores the effects of process combination upon preferable mechanical properties.

Influence of critical variables on tensile characteristics

The effects of critical variables on tensile characteristics of fabricated specimens are depicted in Figures 3 and 4. The graph demonstrates that at medium levels of particle type and size, and lower levels of weight % of the particle, the tensile characteristics of fiber composite are higher. It should also be observed that the “type of particle,” a process variable, affects the tensile properties of the prepared composite specimens. At medium levels of particle type and size, increased filler content reduces the flexibility of the composite, which is a significant finding. As a result, the composite loses its capability to withstand loads and becomes more brittle.

Figure 3. Impact of process variables on tensile strength.

Figure 4. Impact of process variables on tensile modulus.

Influence of critical variables on flexural characteristics

The response of process parameters on the flexural characteristics of the tested fiber composites is shown in Figures 5 and 6. According to the illustration, the flexural properties of fiber composite are higher for medium levels of particle type and size and lower level for particle weight %. Herein, the particle type is a significant parameter that influences the flexural characteristics of the composite samples. The composite’s flexural characteristics increase with the addition of B₄C particle of medium size. The elasticity and brittleness of the composite diminish as particle weight % increases.

Figure 5. Impact of process variables on flexural strength.

Figure 6. Impact of process variables on flexural modulus.

Process variables’ effect on weight gain as a percentage (%)

Figure 7 shows how processing conditions affect the growth in weight % of the samples. The illustration shows that the intermediate level of particle type, greater level of particle size and particle weight % show better results for the water absorption capacity of the fabricated composites.

Figure 7. Impact of process variables on % increase in weight.

Multi-aspect optimization of process parameters

The results of the multi-objective optimization analysis are depicted in Figure 8. The figure shows that the MPI value is higher for medium particle type and particle size and lower percentage weight.

Figure 8. Impact of process variables on GRG.

ANOVA for evaluating the significance of process variables

Table 3 shows the results of a quantitative approach on the considered process variables for selected performance parameters during the production of fiber composites. According to the statistical analysis, the particle weight % is the critical process parameter for the majority of the identified performance metrics. The results of analysis of variance (ANOVA) highlight a correlation between the response and the analysis.

Table 3. Analysis of variance results.

Source	DF		Seq SS	Adj SS			Adj MS	F			P
Analysis of variance for tensile stress
Particle type	2		258.083	313.374			156.687	93.17			0
Particle size (µm)	2		247.887	267.338			133.669	79.48			0
Particle weight %	2		73.729	73.729			36.864	21.92			0.001
Error	8		13.454	13.454			1.682
Total	14		593.153
Analysis of variance for tensile modulus
Source	DF		Seq SS	Adj SS			Adj MS	F			P
Particle type	2		567857	652671			326336	59.19			0
Particle size (µm)	2		451177	467992			233996	42.44			0
Particle weight %	2		121466	121466			60733	11.02			0.005
Error	8		44107	44107			5513
Total	14		1184606
Analysis of variance for flexural stress
Source	DF		Seq SS	Adj SS			Adj MS	F			P
Particle type	2		677.03	773.8			386.9	215.64			0
Particle size (µm)	2		405.73	425.29			212.65	118.52			0
Particle weight %	2		58.09	58.09			29.05	16.19			0.002
Error	8		14.35	14.35			1.79
Total	14		1155.2
Analysis of variance for flexural modulus
Source	DF		Seq SS	Adj SS			Adj MS	F			P
Particle type	2		2961154	3236665			1618332	93.18			0
Particle size (µm)	2		1049675	1084938			542469	31.23			0
Particle weight %	2		116649	116649			58325	3.36			0.087
Error	8		138946	138946			17368
Total	14		4266424
Analysis of variance for % increase in weight
Source	DF		Seq SS	Adj SS			Adj MS	F			P
Particle type	2		0.28805	0.280332			0.140166	47.11			0
Particle size (µm)	2		0.012748	0.013029			0.006514	2.19			0.174
Particle weight %	2		0.051233	0.051233			0.025617	8.61			0.01
Total	8		0.023804	0.023804			0.002976

Multiple regression models to achieve the desired outcome

A regression analysis is performed for the desired output, and the relationship between the desirable performance variables and the model parameters is conferred. For selected performance measures, the evolved linear, quadratic, interaction, and second-order full model equations are depicted in Equations (1–5).

(1) $\begin{array}{rl}& \mathbf{Tensile}\mathbf{Strength}=49.3709+1.44446\mathrm{Particle}\mathrm{Type}+0.267063\mathrm{Particle}\mathrm{Size}-0.446725\\ & \mathrm{Particle}\mathrm{Weight}\mathrm{\%}-8.9639\mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Type}-0.408125\\ & \mathrm{Particle}\mathrm{Type}\ast \mathrm{ParticleSize}+0.525565\mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Weight}\\ & \mathrm{\%}-8.49998\mathrm{ParticleSize}\ast \mathrm{Particle}\mathrm{Size}+0.784\mathrm{Particle}\mathrm{Size}\ast \mathrm{Particle}\\ & \mathrm{Weight}\mathrm{\%}-4.41994\mathrm{Particle}\mathrm{Weight}\mathrm{\%}\ast \mathrm{Particle}\mathrm{Weight}\mathrm{\%}\end{array}$(1)

(2) $\begin{array}{rl}& \mathbf{Tensile}\mathbf{Modulus}=2776.62+97.7659\mathrm{Particle}\mathrm{Type}+28.8161\\ & \mathrm{Particle}\mathrm{Size}-103.734\mathrm{Particle}\mathrm{Weight}\mathrm{\%}-395.039\\ & \mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}-0.2835\mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\\ & \mathrm{Size}-8.68525\mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Weight}\mathrm{\%}-353.481\\ & \mathrm{Particle}\ast \mathrm{Particle}\mathrm{Size}+93.5748\mathrm{Particle}\mathrm{Size}\ast \mathrm{Particle}\\ & \mathrm{Weight}-97.8882\mathrm{Particle}\mathrm{Weight}\mathrm{\%}\ast \mathrm{Particle}\mathrm{Weight}\mathrm{\%}\end{array}$(2)

(3) $\begin{array}{rl}& \mathbf{Flexural}\mathbf{Strength}=81.7552+1.79899\mathrm{Particle}\mathrm{Type}+0.541096\mathrm{Particle}\mathrm{Size}-1.04006\\ & \mathrm{Particle}\mathrm{Weight}\mathrm{\%}-14.2323\mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Type}+0.34095\text{break}\mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\\ & \mathrm{Size}+0.82456\mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Weight}\mathrm{\%}-10.7028\mathrm{Particle}\ast \mathrm{Particle}\mathrm{Size}+0.743873\\ & \mathrm{Particle}\mathrm{Size}\ast \mathrm{Particle}\mathrm{Weight\%}-3.65925\mathrm{Particle}\mathrm{Weight}\mathrm{\%}\ast \mathrm{Particle}\mathrm{Weight}\mathrm{\%}\end{array}$(3)

(4) $\begin{array}{rl}& \mathbf{Flexural}\mathbf{Modulus}=5274.05+166.859\mathrm{Particle}\mathrm{Type}+108.828\\ & \mathrm{Particle}\mathrm{Size}-73.9192\mathrm{Particle}\mathrm{Weight}\mathrm{\%}-903.478\\ & \mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Type}-46.8118\\ & \mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Size}-55.8917\\ & \mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Weight}\mathrm{\%}-517.858\\ & \mathrm{Particle}\mathrm{Size}\ast \mathrm{Particle}\mathrm{Size}-37.9638\mathrm{Particle}\mathrm{Size}\ast \mathrm{Particle}\\ & \mathrm{Weight}\mathrm{\%}-140.548\mathrm{Particle}\mathrm{Weight}\mathrm{\%}\ast \mathrm{Particle}\mathrm{Weight}\mathrm{\%}\end{array}$(4)

(5) $\begin{array}{rl}& \mathbf{Water}\mathbf{Absorption}=0.725689+0.106741\mathrm{Particle}\mathrm{Type}-0.0374882\mathrm{Particle}-0.0783204\\ & \mathrm{Particle}\mathrm{Weight}\mathrm{\%}+0.226356\mathrm{Particle}\ast \mathrm{Particle}\mathrm{Type}-0.0386237\\ & \mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Size}-0.037981\mathrm{Particle}\mathrm{Type}\ast \mathrm{Particle}\mathrm{Weight}\\ & \mathrm{\%}-0.0219923\mathrm{Particle}\mathrm{Size}\ast \mathrm{Particle}\\ & \mathrm{Size}-0.00575975\mathrm{Particle}\mathrm{Size}\ast \mathrm{Particle}\mathrm{Weight}\\ & \mathrm{\%}-0.0241915\mathrm{Particle}\mathrm{Weight}\mathrm{\%}\ast \mathrm{Particle}\mathrm{Weight}\mathrm{\%}\end{array}$(5)

Contour plot analysis on the effect of parameters

Contour plot is utilized to find the location of highest or lowest response predicted from the process. It is especially useful when a maximum or minimum response is anticipated in the contour region inside or near the data range. Contour plots are extremely useful for investigating the combined effects of several control parameters on output characteristics. The contour plots are shown in Figures 9–13.

Figure 9. Contour plot of tensile stress vs. particle type, size, and wt%.

Figure 10. Contour plot of tensile modulus vs. particle type, size, and wt%.

Figure 11. Contour plot of flexural stress vs. particle type, size, and wt%.

Figure 12. Contour plot of flexural modulus vs. particle type, size, and wt%.

Figure 13. Contour plot of water absorption vs. particle type, size and wt%.

Figures 9 and 10 present the contour plots showing the combined effects of the particle type, particle size, and particle weight % on the tensile characteristics. According to the plots, the tensile stress and modulus are maximum at the medium particle type and particle size level.

The contour plots, presented in Figures 11 and 12, are extracted to study the influence of particle type, particle size, and weight % on flexural properties. It is found from the plots that the maximum flexural strength and modulus are achieved for the amalgamations of the medium level of particle type and particle size and low level of particle weight %.

The water absorption capacity of the prepared composites is analyzed, which is shown in Figure 13. During the study on the influence of process parameters on water absorption, it is found that water absorption capacity is improved when the particle type is medium, the particle size is large, and the particle weight % is high.

Evolution of ANFIS model for jute fiber composites

The ANFIS framework is evolved using the MATLAB toolbox to predict the intended multi-performance indicator. This work’s ANFIS model has one output parameter and five input parameters. The built model is trained and thereafter utilized to imagine the required measurements. The training details involved in this study are mentioned in Table 4.

Table 4. Training parameters for ANFIS model.

Number of membership functions available at each input level	3 3 3 3 3
Type of input membership function	trimf
Type of output membership function	constant
Method of optimization	Back propagation
Error tolerance	0
No. of epochs	500

The 243 rules in the ANFIS model were generated with “trimf,“ through the input. Grey theory is one of the conventional approaches engaged in attaining the Grey Relational Coefficient and is given as input data. For further processing, the outcomes that are evaluated are normalized. The ANFIS editor for jute fiber composites is shown in Figures 14 and 15. The same is involved in estimating the chosen MPIs (ANFIS-GRG). Figures 16 and 17 represent the FIS editor and rule viewer, respectively.

Figure 14. ANFIS editor.

Figure 15. ANFIS model structure for developed composites.

Figure 16. FIS editor.

Figure 17. ANFIS Rule viewer.

Interpretation on ANFIS analysis to envisage GRG

Typical surface plots for determining process variables’ effect on jute fiber composites using ANFIS-GRG are shown in Figures 18–21. The impact of GRC of flexural strength with GRC of flexural modulus, tensile strength, tensile modulus, and increase in weight % based on ANFIS-GRG is studied. It is evident from Figure 18 that the mid level of Grey relational coefficient of flexural strength and flexural modulus is better and improved from ANFIS-GRG. Figure 19 depicts that similar findings are made at intermediate level of tensile modulus. It is noted from Figures 20–21 showing ANFIS-GRG that a higher level of Grey relational coefficient of flexural strength and a midlevel of tensile strength are finer and improved and likewise, the higher level of % increase in weight.

Figure 18. Influence of GRC flexural strength and GRC flexural modulus on ANFIS-GRG.

Figure 19. Influence of GRC flexural strength and GRC tensile modulus on ANFIS-GRG.

Figure 20. Influence of GRC flexural strength and GRC tensile modulus on ANFIS-GRG.

Figure 21. Influence of GRC flexural strength and GRC of % weight on ANFIS-GRG.

Analysis on performance of the developed ANFIS model

Here, $E_i$ is the observed value, $P_i$ is the predicted value from the model, $\bar{E}$ is the average of the observed values, and n is the number of experiments. The error values obtained from Equations (6–13) are represented in Table 5. Performance analysis highlighting minimum error is shown in Figure 22.

Figure 22. Various error values of developed ANFIS model.

Table 5. Performance analysis of Grey-ANFIS model error.

Model-ANFIS	Error
Mean absolute error (MAE)	4.6000e-04
Mean squared error (MSE)	1.6140e-06
Root mean squared error (RMSE)	0.0013
Mean absolute relative error (MARE)	4.9824e-04
Mean squared relative error (MSRE)	1.8943e-06
Root mean squared relative error (RMSRE)	0.0014
Mean absolute percentage error (MAPE)	0.0498
Mean squared percentage error (MSPE)	0.0189

The error values obtained from Equations (6–13) are represented in Table 5.

(6) $\mathrm{Mean}\mathrm{Absolute}\mathrm{Error}:\mathit{MAE}=\frac{{\sum }_{i=1}^n\left|E_i-P_i\right|}{\mathit{n}}$(6)

(7) $\mathrm{Mean}\mathrm{squared}\mathrm{error}:\mathit{MSE}=\frac{{\sum }_{i=1}^n{\left(E_i-P_i\right)}^2}{\mathit{n}}$(7)

(8) $\mathrm{Root}\mathrm{mean}\mathrm{squared}\mathrm{error}:\mathit{RMSE}=\sqrt{\frac{{\sum }_{i=1}^n{\left(E_i-P_i\right)}^2}{\mathit{n}}}$(8)

(9) $\mathrm{Mean}\mathrm{absolute}\mathrm{relative}\mathrm{error}:\mathit{MARE}=\frac{{\sum }_{i=1}^n\left|\frac{\left(E_i-P_i\right)}{E_i}\right|}{\mathit{n}}$(9)

(10) $\mathrm{Mean}\mathrm{squared}\mathrm{relative}\mathrm{error}:\mathit{MSRE}=\frac{{\sum }_{i=1}^n\left|\frac{\left(E_i-P_i\right)}{E_i}\right|}{\mathit{n}}$(10)

(11) $\mathrm{Root}\mathrm{mean}\mathrm{squared}\mathrm{relative}\mathrm{error}:\mathit{RMSRE}=\sqrt{\frac{{\sum }_{i=1}^n{\left(\frac{E_i-P_i}{E_i}\right)}^2}{\mathit{n}}}$(11)

(12) $\mathrm{Mean}\mathrm{absolute}\mathrm{percentage}\mathrm{error}:\mathit{MAPE}=\frac{{\sum }_{i=1}^n\left|\frac{\left(E_i-P_i\right)}{E_i}\right|}{\mathit{n}}\ast 100$(12)

(13) $\mathrm{Mean}\mathrm{squared}\mathrm{percentage}\mathrm{error}:\mathit{MSPE}=\frac{{\sum }_{i=1}^n{\left(\frac{E_i-P_i}{E_i}\ast 100\right)}^2}{\mathit{n}}$(13)

Analyzed model efficiency

The model’s efficiency is also analyzed using Equations (14) and (15), and the values obtained from this analysis are depicted in Table 6.

(14) $\mathrm{Correlation}\mathrm{coefficient}:\mathit{R}=\frac{n\ast \left({\sum }_{i=1}^n{E}_i{P}_i\right)-\left(\left({\sum }_{i=1}^n{E}_i\right)\ast \left({\sum }_{i=1}^n{P}_i\right)\right)}{\sqrt{\left(\mathit{n}\ast {\sum }_{i=1}^n{E_i}^2-{\left({\sum }_{i=1}^n{E}_i\right)}^2\right)}\ast \sqrt{\left(\mathit{n}\ast {\sum }_{i=1}^n{P_i}^2-{\left({\sum }_{i=1}^n{P}_i\right)}^2\right)}}$(14)

(15) $\mathrm{Nash}\mathrm{Sutcliffe}\mathrm{efficiency}\mathrm{coefficient}\left(\mathrm{NSE}\right)\mathit{NSE}=1-\frac{{\sum }_{i=1}^n{\left(E_i-P_i\right)}^2}{{\sum }_{i=1}^n{\left(E_i-\bar{E}\right)}^2}$(15)

Table 6. Performance analysis of Grey-ANFIS model efficiency.

Model – ANFIS	Efficiency
Correlation coefficient (R)	0.99999
Nash Sutcliffe efficiency coefficient (NSE)	0.999956925

The assessment of the fabricated composites is evaluated using the proposed ANFIS model, and the results are compared with those obtained through experimentation. A significant relation was found between actual and desired results. Equations (14)(14) $\mathrm{Correlation}\mathrm{coefficient}:\mathit{R}=\frac{n\ast \left({\sum }_{i=1}^n{E}_i{P}_i\right)-\left(\left({\sum }_{i=1}^n{E}_i\right)\ast \left({\sum }_{i=1}^n{P}_i\right)\right)}{\sqrt{\left(\mathit{n}\ast {\sum }_{i=1}^n{E_i}^2-{\left({\sum }_{i=1}^n{E}_i\right)}^2\right)}\ast \sqrt{\left(\mathit{n}\ast {\sum }_{i=1}^n{P_i}^2-{\left({\sum }_{i=1}^n{P}_i\right)}^2\right)}}$(14) and (15(15) $\mathrm{Nash}\mathrm{Sutcliffe}\mathrm{efficiency}\mathrm{coefficient}\left(\mathrm{NSE}\right)\mathit{NSE}=1-\frac{{\sum }_{i=1}^n{\left(E_i-P_i\right)}^2}{{\sum }_{i=1}^n{\left(E_i-\bar{E}\right)}^2}$(15) ) are used to assess the performance of the developed model as shown in Table 6.

The foreseen values of jute fiber composites obtained using the created model are compared with the corresponding experimental results. It was found that a considerable relationship exists between experimental findings and the values obtained from the considered model. A visual comparison of the observed values between GRG and ANFIS-GRG is shown in Figure 23. The comparison demonstrated a close association between the tested and developed models.

Figure 23. Comparison of actual GRG and ANFIS predicted GRG.

Conclusions

BBD technique is used in this work to obtain possible combinations of operating parameters. The performance measures used in this study are tensile characteristics, flexural properties, and increase in weight %. A three-level/three-parameter model is adopted. The compression molding machine has been employed to fabricate the composites. The mechanical characteristics of the fabricated composites are investigated. ANOVA is used to identify the relevance of individual process variables. According to the ANOVA assessment, particle type is the influential process variable. Mathematical correlations are established, and the R² values of the produced models are within limits, indicating that the predicted model is suitable for future assessment. The Grey system theory is considered to evaluate the multi-performance index and the GRC values generated by this study and to construct a hybrid ANFIS-GRG model. The model’s competency is demonstrated by its performance evaluation.

Highlights

• Box–Behnken Design approach is used for designing the experiments.

• Compression molding machine is used to fabricate the composites.

• ANOVA is used to identify the relevance of individual process variables.

• Particle type is the most influential process variable.

• Hybrid Intelligent Decision-Making Tool-Grey ANFIS is used to predict the desired output measure.

Disclosure statement

No potential conflict of interest was reported by the author(s).