# A Novel Performance Prediction Model for the Machining Process Based on the Interval Type-2 Fuzzy Neural Network

Authors: Wenwen Tian; Fei Zhao; Zheng Sun; Suiyan Shang; Xuesong Mei; Guangde Chen
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

The prediction model is the most important part of the virtual metrology system. Predicting the performance of the machining process has been widely applied in manufacturing, which can reduce costs and improve efficiency compared with the manual operation. In this paper, a novel performance prediction model for the machining process is proposed based on the interval type-2 fuzzy neural network. The interval type-2 fuzzy logic system with a complete rule base, type-reduction, and defuzzified output is simplified by the BMM method to meet the requirements of the prediction. The proposed prediction model is trained using a gradient-based optimization algorithm. To evaluate the performance of the proposed approach, it is applied to wire electrical discharge turning process for predicting material removal rate and surface roughness with a published dataset. The results show that the proposed method is an effective scheme in the studied cases.


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## Body

## 1. Introduction

The main objective in the machining process is the improvement of the product quality and productivity at the same time. In order to ensure that product quality meets the requirements, the machining quality inspection of multiple key parts is an essential link in the production process. However, it is almost impossible to perform the total inspection for all parts in real manufacturing processes because such an inspection takes remarkable time and imposes an undeniable delay in the quality control. Although the quality in this method is guaranteed, the continuity of the production is adversely affected. In the context of intelligent manufacturing, finding novel intelligent detection methods is highly required. Reviewing the literature indicates that several schemes have been proposed in this area so far. More specifically, the virtual metrology technology is an effective scheme to perform the quality process monitoring. Investigation and implementation of the virtual metrology have increasingly attracted attentions from both academia and industry. Studies show that this method can effectively reduce the detection time and achieve the total inspection of the product quality. The virtual metrology has been widely applied in diverse high-tech industries, including the semiconductor industry [1], TFT-LCD industry [2], and the solar-cell industry [3]. Moreover, the virtual metrology technology has been applied for processing the CNC machine tool and reasonable results have been achieved accordingly [4–7].In recent years, different modelling techniques in prediction of machining performances have been proposed for various processes. Because of the causal relationship between machining process and performances, machining process parameters and performances are often used as input and output for modelling, respectively. Due to complexity and uncertainty of the machining processes, soft computing techniques are being preferred to physics-based models for predicting the performance of the machining processes and optimizing them. A review of application of soft computing techniques in machining performance prediction and optimization has been presented [8]. Machining is the core process in the manufacturing industry, which intends to achieve either a minimum production cost, maximum production rate, or an optimum combination of both to obtain a reasonable production quality through the machining [9]. Meanwhile, the prediction model has attracted interests in the industrial production area. Such interests are especially more pronounced in estimating the machining quality [10], remaining useful life [11, 12], energy consumption [13], and tool wear [14].Depending on the types of data and the information characteristics of the interested systems, prognosis techniques can be classified into three main categories, including the physics-based, data-driven, and model-based techniques [15]. Many machining monitoring systems based on data-driven artificial intelligence process models have been developed in the past for the optimizing, predicting, or controlling diverse machining processes. Baruah and Chinnam [16] proposed a novel method employing hidden Markov models to conduct both diagnostic and prognostic activities for metal cutting tools. To ensure the stability and quality of machining processes in multistage machining processes (MMPs), a real-time quality monitoring and predicting model was proposed based on error propagation networks [17]. In order to perform the online prediction of the workpiece deformation during the machining, a purely data-driven sparse Bayesian learning-based method was developed [18], which predicts the future deformation of the workpiece through the historical displacement information.Abnormal machining condition causes quality loss in the finished parts. To this end, Lu et al. [19] proposed a hybrid condition monitoring approach for detecting multiple abnormal conditions in complicated machining process through the deep forest and multiprocess information fusion. Furthermore, Chiu and Lee [20] proposed an intelligent prediction system of milling accuracy and surface quality for CNC machining parameters based on the adaptive neurofuzzy inference systems (ANFIS). They showed that this predicted system can help user to achieve the desired product quality and the machining productivity.Interval type-2 fuzzy neural networks (IT2FNN) combine the reasoning ability of the interval type-2 fuzzy logic system (IT2FLS) and the self-learning ability of the neural network so that it is a powerful scheme in dealing with the uncertain and nonlinear problems. Although the IT2FNN method has been widely applied in diverse fields, it has been rarely applied in the machining. Type-2 fuzzy logic estimation provides a possibility to show uncertainties in the manufacturing process and can automate the process monitoring, which is of significant importance in maintaining the high quality production. Chen and Vachtsevanos [21] proposed an IT2FNN scheme to perform the multistep-ahead condition prediction in faulty bearings. Its results show a better prediction accuracy compared with the ANFIS method. Since the unforeseeable changes may occur frequently in the dynamic machining environment, Tseng et al. [22] proposed a new methodology to predict a system output by applying a data mining technique and a hybrid type-2 fuzzy system in CNC turning operations. Although the IT2FNN method has more advantages in processing uncertainty, incomplete, and imprecise information compared to its type-1 counterparts, a large number of parameters should be tuned in the IT2FNN method. The structure design of the IT2FNN method is the key step that affects the overall performance. Wu and Mendel [23] recommended some representative starting choices for designing an IT2 fuzzy system. The most challenging questions here are about output processing and type-reduction methods. It is worth noting that many type-reduction methods have been proposed so far [24]. In the present study, it is intended to apply the Begian–Melek–Mendel (BMM) [25] method to compute the crisp output of the IT2FLS. It introduces a closed form for the inference, which eliminates the need to use a type reducer or any complicated defuzzification techniques.Wire electrical discharge machining (WEDM) is a widely accepted material removal process for components with intricate shapes and profiles. Moreover, wire electrical discharge turning (WEDT) is an emerging area and it can be used for generating cylindrical forms on difficult-to-machine materials by adding a rotary axes on the WEDM. The WEDT process can be modelled through the artificial neural network with feedforward backpropagation algorithm and ANFIS [26]. Sarkheyli et al. [27] proposed a new hybrid technique anchored in the ANFIS and modified genetic algorithm (MGA) to model the correlation between machining parameters and multi-performances, then applied it to the WEDM for predicting material removal rate (MRR) and surface roughness (Ra).In the present study, it is intended to present a novel prediction model based on the simplified interval type-2 fuzzy neural network for predicting the MRR and Ra in the WEDT, which are important for increasing the productivity and quality of the products. To this end, the IT2FNN method with a complete rule base and the BMM method will be applied to simplify the type-reduction and defuzzified output of the IT2FLS.The remainder of this article is organized as follows. Structure of the interval type-2 fuzzy neural network and parameters of the learning algorithm are presented in Section2. Then, case study and discussion are presented in Section 3. Finally, main conclusions are presented in Section 4.


## 2. Prediction Model Based on the Interval Type-2 Fuzzy Neural Network

### 2.1. Structure of the Interval Type-2 Fuzzy Neural Network

In this section, it is intended to introduce the structure of the simplified IT2FNN method. With no loss of generality, multiple-input multiple-output (MIMO) fuzzy systems can be decomposed into a series of multiple-input single-output (MISO) fuzzy systems. Figure1 illustrates the structure of a MISO IT2FNN scheme with six layers. This six-layered network forms an interval type-2 fuzzy logic system whose consequent part is a linear combination of input variables.Figure 1
Structure of the interval type-2 fuzzy neural network.In the present study, an interval type-2 fuzzy logic system is considered with a complete rule base, and the consequent parts of the rules are of the Takagi–Sugeno–Kang (TSK) type. In this case, thej1,j2,…,jnth rule can be expressed as follows:(1)Rj1,j2,…,jn:ifx1isA˜1j1andx2isA˜2j2and…andxnisA˜njn,thenwj1,j2,…,jnisbj1,j2,…,jn+∑i=1naij1,j2,…,jnxi,where x1,x2,…,xn are the input variables, wj1,j2,…,jn is the output variable, and A˜iji is the interval type-2 fuzzy set for the jith rule with the ith input. Parameters in the consequent part of rules are aij1,j2,…,jn and bj1,j2,…,jni=1,2,…,n,ji∈1,2,…,m.The specific expression of each layer is discussed as follows.(i)
Layer 1 (the input layer): this layer containsn neurons. Each node is a crisp input variable. It should be indicated that there are no weights to be adjusted in this layer.(ii)
Layer 2 (the membership function layer): in this article, the Gaussian primary membership function (MF) with uncertain standard deviation and fixed center value is applied in the interval type-2 fuzzy setA˜iji. Subsequently, the footprint of uncertainty (FOU) of this MF can be expressed as a bounded interval in terms of an upper MF μ¯A˜iji and a lower MF μ¯A˜iji, as shown in Figure 2. mn shows the total number of neurons in the MF layer. Each node utilizes an interval type-2 MF to perform the fuzzification operation:(2)μA˜ijixi=exp−12xi−cijiσiji2≡Nciji,σiji;xi,whereμA˜iji is the jith MF with the ith input, while cijiand σiji denote the center and widths of the MF, respectively.Then, the output of each node can be represented as an intervalμ¯A˜iji,μ¯A˜iji, with the upper and lower bounds defined as follows:(3)μ¯A˜ijixi=exp−12xi−cijiσ¯iji2≡Nciji,σ¯iji;xi,μ¯A˜ijixi=exp−12xi−cijiσ¯iji2≡Nciji,σ¯iji;xi.(iii)
Layer 3 (the rule layer): this layer containsM=mn neurons. Each node in this layer represents the antecedent part of a fuzzy rule computing the firing strength. This layer performs the product operation to obtain the fuzzy firing strength. Since the MF layer has an interval value of μ¯A˜iji,μ¯A˜iji, the firing strength Fj1,j2,…,jn for the rule j1,j2,…,jnth is considered as an interval type-1 fuzzy set f¯j1,j2,…,jn,f¯j1,j2,…,jn, ji∈1,2,…,m so that it can be computed through the following expression:(4)f¯j1,j2,…,jnxi=∏i=1nμ¯A˜ijixi,f¯j1,j2,…,jnxi=∏i=1nμ¯A˜ijixi.(iv)
Layer 4 (the normalization layer): the number of neurons in this layer is alsoM=mn, which is the same as the number of neurons in layer 3. The outputs of this layer are called normalized firing strengths. The normalization follows the formula below:(5)f^¯j1,j2,…,jn=f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,f¯^j1,j2,…,jn=f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn.(v)
Layer 5 (the type-reduction layer): the outputs of this layer can be obtained by the firing nodes and the connecting weight vectors, which are defined in the form below:(6)yl=∑j1=1m∑j2=1m…∑jn=1mf^¯j1,j2,…,jnwj1,j2,…,jn=∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jnwj1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,yr=∑j1=1m∑j2=1m…∑jn=1mf¯^j1,j2,…,jnwj1,j2,…,jn=∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jnwj1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,wherewj1,j2,…,jn is a linear combination of input variables, which can be expressed as follows:(7)wj1,j2,…,jn=bj1,j2,…,jn+∑i=1naij1,j2,…,jnxi.(vi)
Layer 6 (the output layer): each node in this layer computes the output variable through the defuzzification operation. The output value of this layer can be expressed in the form below:(8)y=δyl+1−δyr,where the design factor δ∈0,1 adjusts the proportion of the upper to the lower bound, depending on the certainty of the system. Therefore, there is no need to apply the Karnik–Mendel (KM) iterative procedure to find the end points [24] and reduce the computational complexity of the type-2 fuzzy logic system for large number of rules.Figure 2
Distribution of the interval type-2 membership function with uncertain standard deviation.


### 2.2. Learning Algorithm

In this section, it is intended to use the gradient descent method to derive the parameter learning of the IT2FNN system. For clarification, a single-output system is studied to minimize the objective error. The error function is defined as follows:(9)Et=12et2=12yt−yat2,where yt and yat are the model output and the actual output, respectively. Applying the gradient descent method, the updated law of parameters can be expressed as follows:(10)Wt+1=Wt+ΔWt,where ΔW=−η⋅∂Et/∂W and 0≤δ≤1 is the learning rate. Moreover, adjustable parameters are represented as W=w¯,w¯,wab,δT, where w¯,w¯,wab,andδ are the parameters of upper membership functions, lower membership functions, consequent weights, and the proportion factor, respectively. The term with the partial derivative can be expressed as follows:(11)∂Et∂W=∂Et∂yt⋅∂yt∂W=yt−yat⋅∂yt∂W=et⋅∂yt∂W.Therefore, the parameters of the updated law equation can be rewritten as follows:(12)Wt+1=Wt−η⋅et⋅∂yt∂W.Then, consequent parameters used in the layer 4 are tuned by the following equations:(13)aij1,j2,…,jnt+1=aij1,j2,…,jnt−η⋅∂E∂aij1,j2,…,jn,where(14)∂E∂aij1,j2,…,jn=∂E∂y∂y∂wj1,j2,…,jn∂wj1,j2,…,jn∂aij1,j2,…,jn=et⋅δ⋅f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn−1−δ⋅f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn⋅xit,bj1,j2,…,jnt+1=bj1,j2,…,jnt−η⋅∂E∂bj1,j2,…,jn,∂E∂bj1,j2,…,jn=∂E∂y∂y∂wj1,j2,…,jn∂wj1,j2,…,jn∂bj1,j2,…,jn=et⋅δ⋅f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn−1−δ⋅f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn.Antecedent parameters used in layer 2 are tuned as follows:(15)cijit+1=cijit−η⋅∂E∂ciji,where(16)∂E∂ciji=∑j1=1m∑j2=1m…∑jn=1m∂E∂y∂y∂yl∂yl∂f¯j1,j2,…,jn∂f¯j1,j2,…,jn∂μ¯A˜iji∂μ¯A˜iji∂ciji+∂y∂yr∂yr∂f¯j1,j2,…,jn∂f¯j1,j2,…,jn∂μ¯A˜iji∂μ¯A˜iji∂ciji,σ¯ijit+1=σ¯ijit−η⋅∂E∂σ¯iji,∂E∂σ¯iji=∑j1=1m∑j2=1m⋯∑jn=1m∂E∂y∂y∂yl∂yl∂f¯j1,j2,⋅⋅⋅,jn∂f¯j1,j2,⋅⋅⋅,jn∂μ¯A˜iji∂μ¯A˜iji∂σ¯iji,σ¯ijit+1=σ¯ijit−η⋅∂E∂σ¯iji,∂E∂σ¯iji=∑j1=1m∑j2=1m⋯∑jn=1m∂E∂y∂y∂yr∂yr∂f¯j1,j2,⋅⋅⋅,jn∂f¯j1,j2,⋅⋅⋅,jn∂μ¯A˜iji∂μ¯A˜iji∂σ¯iji.Partial derivatives in the foregoing expressions can be expressed as follows:(17)∂E∂y=yt−yat=et,∂y∂yl=δ,∂y∂yr=1−δ,∂yl∂f¯j1,j2,…,jn=wj1,j2,…,jn−yl∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,∂yr∂f¯j1,j2,…,jn=wj1,j2,…,jn−yr∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,∂f¯j1,j2,…,jn∂μ¯A˜iji=∏k=1k≠inμ¯A˜kjkxk,∂f¯j1,j2,…,jn∂μ¯A˜iji=∏k=1k≠inμ¯A˜kjkxk,∂μ¯A˜iji∂ciji=exp−12xi−cijiσ¯iji2⋅xi−cijiσ¯iji2,∂μ¯A˜iji∂ciji=exp−12xi−cijiσ¯iji2⋅xi−cijiσ¯iji2,∂μ¯A˜iji∂σ¯iji=exp−12xi−cijiσ¯iji2⋅xi−ciji2σ¯iji3,∂μ¯A˜iji∂σ¯iji=exp−12xi−cijiσ¯iji2⋅xi−ciji2σ¯iji3.It is worth noting that the parameterδ adjusts the lower or upper portions in the final output. Subsequently, the proportion factor can be tuned as follows:(18)δt+1=δt−η⋅∂E∂δ,where(19)∂E∂δ=∂E∂y∂y∂δ=et⋅∑j=1Mf¯jwj∑j=1Mf¯j−∑j=1Mf¯jwj∑j=1Mf¯j.An important issue for each learning algorithm is the convergence. The convergence of the gradient descent method depends on the selection of the initial value for the learning rate. The convergence is guaranteed when the following inequality is satisfied [28]:(20)0<ηt<2maxt∂yt/∂W2.


## 2.1. Structure of the Interval Type-2 Fuzzy Neural Network

In this section, it is intended to introduce the structure of the simplified IT2FNN method. With no loss of generality, multiple-input multiple-output (MIMO) fuzzy systems can be decomposed into a series of multiple-input single-output (MISO) fuzzy systems. Figure1 illustrates the structure of a MISO IT2FNN scheme with six layers. This six-layered network forms an interval type-2 fuzzy logic system whose consequent part is a linear combination of input variables.Figure 1
Structure of the interval type-2 fuzzy neural network.In the present study, an interval type-2 fuzzy logic system is considered with a complete rule base, and the consequent parts of the rules are of the Takagi–Sugeno–Kang (TSK) type. In this case, thej1,j2,…,jnth rule can be expressed as follows:(1)Rj1,j2,…,jn:ifx1isA˜1j1andx2isA˜2j2and…andxnisA˜njn,thenwj1,j2,…,jnisbj1,j2,…,jn+∑i=1naij1,j2,…,jnxi,where x1,x2,…,xn are the input variables, wj1,j2,…,jn is the output variable, and A˜iji is the interval type-2 fuzzy set for the jith rule with the ith input. Parameters in the consequent part of rules are aij1,j2,…,jn and bj1,j2,…,jni=1,2,…,n,ji∈1,2,…,m.The specific expression of each layer is discussed as follows.(i)
Layer 1 (the input layer): this layer containsn neurons. Each node is a crisp input variable. It should be indicated that there are no weights to be adjusted in this layer.(ii)
Layer 2 (the membership function layer): in this article, the Gaussian primary membership function (MF) with uncertain standard deviation and fixed center value is applied in the interval type-2 fuzzy setA˜iji. Subsequently, the footprint of uncertainty (FOU) of this MF can be expressed as a bounded interval in terms of an upper MF μ¯A˜iji and a lower MF μ¯A˜iji, as shown in Figure 2. mn shows the total number of neurons in the MF layer. Each node utilizes an interval type-2 MF to perform the fuzzification operation:(2)μA˜ijixi=exp−12xi−cijiσiji2≡Nciji,σiji;xi,whereμA˜iji is the jith MF with the ith input, while cijiand σiji denote the center and widths of the MF, respectively.Then, the output of each node can be represented as an intervalμ¯A˜iji,μ¯A˜iji, with the upper and lower bounds defined as follows:(3)μ¯A˜ijixi=exp−12xi−cijiσ¯iji2≡Nciji,σ¯iji;xi,μ¯A˜ijixi=exp−12xi−cijiσ¯iji2≡Nciji,σ¯iji;xi.(iii)
Layer 3 (the rule layer): this layer containsM=mn neurons. Each node in this layer represents the antecedent part of a fuzzy rule computing the firing strength. This layer performs the product operation to obtain the fuzzy firing strength. Since the MF layer has an interval value of μ¯A˜iji,μ¯A˜iji, the firing strength Fj1,j2,…,jn for the rule j1,j2,…,jnth is considered as an interval type-1 fuzzy set f¯j1,j2,…,jn,f¯j1,j2,…,jn, ji∈1,2,…,m so that it can be computed through the following expression:(4)f¯j1,j2,…,jnxi=∏i=1nμ¯A˜ijixi,f¯j1,j2,…,jnxi=∏i=1nμ¯A˜ijixi.(iv)
Layer 4 (the normalization layer): the number of neurons in this layer is alsoM=mn, which is the same as the number of neurons in layer 3. The outputs of this layer are called normalized firing strengths. The normalization follows the formula below:(5)f^¯j1,j2,…,jn=f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,f¯^j1,j2,…,jn=f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn.(v)
Layer 5 (the type-reduction layer): the outputs of this layer can be obtained by the firing nodes and the connecting weight vectors, which are defined in the form below:(6)yl=∑j1=1m∑j2=1m…∑jn=1mf^¯j1,j2,…,jnwj1,j2,…,jn=∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jnwj1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,yr=∑j1=1m∑j2=1m…∑jn=1mf¯^j1,j2,…,jnwj1,j2,…,jn=∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jnwj1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,wherewj1,j2,…,jn is a linear combination of input variables, which can be expressed as follows:(7)wj1,j2,…,jn=bj1,j2,…,jn+∑i=1naij1,j2,…,jnxi.(vi)
Layer 6 (the output layer): each node in this layer computes the output variable through the defuzzification operation. The output value of this layer can be expressed in the form below:(8)y=δyl+1−δyr,where the design factor δ∈0,1 adjusts the proportion of the upper to the lower bound, depending on the certainty of the system. Therefore, there is no need to apply the Karnik–Mendel (KM) iterative procedure to find the end points [24] and reduce the computational complexity of the type-2 fuzzy logic system for large number of rules.Figure 2
Distribution of the interval type-2 membership function with uncertain standard deviation.


## 2.2. Learning Algorithm

In this section, it is intended to use the gradient descent method to derive the parameter learning of the IT2FNN system. For clarification, a single-output system is studied to minimize the objective error. The error function is defined as follows:(9)Et=12et2=12yt−yat2,where yt and yat are the model output and the actual output, respectively. Applying the gradient descent method, the updated law of parameters can be expressed as follows:(10)Wt+1=Wt+ΔWt,where ΔW=−η⋅∂Et/∂W and 0≤δ≤1 is the learning rate. Moreover, adjustable parameters are represented as W=w¯,w¯,wab,δT, where w¯,w¯,wab,andδ are the parameters of upper membership functions, lower membership functions, consequent weights, and the proportion factor, respectively. The term with the partial derivative can be expressed as follows:(11)∂Et∂W=∂Et∂yt⋅∂yt∂W=yt−yat⋅∂yt∂W=et⋅∂yt∂W.Therefore, the parameters of the updated law equation can be rewritten as follows:(12)Wt+1=Wt−η⋅et⋅∂yt∂W.Then, consequent parameters used in the layer 4 are tuned by the following equations:(13)aij1,j2,…,jnt+1=aij1,j2,…,jnt−η⋅∂E∂aij1,j2,…,jn,where(14)∂E∂aij1,j2,…,jn=∂E∂y∂y∂wj1,j2,…,jn∂wj1,j2,…,jn∂aij1,j2,…,jn=et⋅δ⋅f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn−1−δ⋅f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn⋅xit,bj1,j2,…,jnt+1=bj1,j2,…,jnt−η⋅∂E∂bj1,j2,…,jn,∂E∂bj1,j2,…,jn=∂E∂y∂y∂wj1,j2,…,jn∂wj1,j2,…,jn∂bj1,j2,…,jn=et⋅δ⋅f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn−1−δ⋅f¯j1,j2,…,jn∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn.Antecedent parameters used in layer 2 are tuned as follows:(15)cijit+1=cijit−η⋅∂E∂ciji,where(16)∂E∂ciji=∑j1=1m∑j2=1m…∑jn=1m∂E∂y∂y∂yl∂yl∂f¯j1,j2,…,jn∂f¯j1,j2,…,jn∂μ¯A˜iji∂μ¯A˜iji∂ciji+∂y∂yr∂yr∂f¯j1,j2,…,jn∂f¯j1,j2,…,jn∂μ¯A˜iji∂μ¯A˜iji∂ciji,σ¯ijit+1=σ¯ijit−η⋅∂E∂σ¯iji,∂E∂σ¯iji=∑j1=1m∑j2=1m⋯∑jn=1m∂E∂y∂y∂yl∂yl∂f¯j1,j2,⋅⋅⋅,jn∂f¯j1,j2,⋅⋅⋅,jn∂μ¯A˜iji∂μ¯A˜iji∂σ¯iji,σ¯ijit+1=σ¯ijit−η⋅∂E∂σ¯iji,∂E∂σ¯iji=∑j1=1m∑j2=1m⋯∑jn=1m∂E∂y∂y∂yr∂yr∂f¯j1,j2,⋅⋅⋅,jn∂f¯j1,j2,⋅⋅⋅,jn∂μ¯A˜iji∂μ¯A˜iji∂σ¯iji.Partial derivatives in the foregoing expressions can be expressed as follows:(17)∂E∂y=yt−yat=et,∂y∂yl=δ,∂y∂yr=1−δ,∂yl∂f¯j1,j2,…,jn=wj1,j2,…,jn−yl∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,∂yr∂f¯j1,j2,…,jn=wj1,j2,…,jn−yr∑j1=1m∑j2=1m…∑jn=1mf¯j1,j2,…,jn,∂f¯j1,j2,…,jn∂μ¯A˜iji=∏k=1k≠inμ¯A˜kjkxk,∂f¯j1,j2,…,jn∂μ¯A˜iji=∏k=1k≠inμ¯A˜kjkxk,∂μ¯A˜iji∂ciji=exp−12xi−cijiσ¯iji2⋅xi−cijiσ¯iji2,∂μ¯A˜iji∂ciji=exp−12xi−cijiσ¯iji2⋅xi−cijiσ¯iji2,∂μ¯A˜iji∂σ¯iji=exp−12xi−cijiσ¯iji2⋅xi−ciji2σ¯iji3,∂μ¯A˜iji∂σ¯iji=exp−12xi−cijiσ¯iji2⋅xi−ciji2σ¯iji3.It is worth noting that the parameterδ adjusts the lower or upper portions in the final output. Subsequently, the proportion factor can be tuned as follows:(18)δt+1=δt−η⋅∂E∂δ,where(19)∂E∂δ=∂E∂y∂y∂δ=et⋅∑j=1Mf¯jwj∑j=1Mf¯j−∑j=1Mf¯jwj∑j=1Mf¯j.An important issue for each learning algorithm is the convergence. The convergence of the gradient descent method depends on the selection of the initial value for the learning rate. The convergence is guaranteed when the following inequality is satisfied [28]:(20)0<ηt<2maxt∂yt/∂W2.


## 3. Case Study and Discussion

### 3.1. Designing the Experiment

In this section, it is intended to evaluate the performance of the proposed method in predicting the MRR and Ra in the WEDT. The experimental dataset in this paper is published in [26]. The selected material for conducting the experiments is AISI D3 (DIN X210Cr12) tool steel with the Brinell hardness of about 212–248 HB. The diameter of the workpieces and the designed depth of cut are set to 10 mm and 0.1 mm, respectively. Pulse off-time, spark gap, servo feed, rotational speed, and flushing pressure are selected, as shown in Table 1, to analyse their impact on the MRR and Ra in the WEDT.Table 1
Datasets for designing the experiment [26].
No.Pulse off-time (μs)Spark gap (μm)Servo feed (level)Rotational speed (rpm)Flushing pressure (bar)MRR (mm3/min)Ra (μm)130303303.2671.242.396230505703.2672.213.7563308081003.2672.604.1344343051003.2671.733.172534508303.2673.785.009634803703.2671.452.827742308703.2671.502.8998425031003.2670.952.116942805303.2671.683.1331030303301.8931.162.3111130505701.8932.203.53912308081001.8932.474.00313343051001.8931.402.6941434508301.8933.724.8971534803701.8931.372.5021642308701.8931.462.89517425031001.8930.822.0691842805301.8931.653.1241930303301.2631.102.2642030505701.2631.482.86021308081001.2632.223.79422343051001.2631.342.5062334508301.2633.204.5762434803701.2631.352.5202542308701.2631.372.58826425031001.2630.782.0482742805301.2631.542.9872830508703.2672.804.2812934805501.8931.442.7443042503501.2630.982.149


### 3.2. Designing Factors of the Model Architecture

MRR and Ra are separately predicted through the proposed IT2FNN scheme. In this regard, the IT2FNN architecture factors are presented in Table2. The sample data should be initially normalized prior to the training and testing to remove the effect of different dimension scales on the results. The normalization process is defined as follows:(21)x¯i=xi−xminxmax−xmin,where xi and x¯i denote the sampling data and the corresponding normalized data, respectively. Moreover, xmax and xmin are the maximum and minimum values of the sampling data, respectively.Table 2
Design factors in the IT2FNN architecture.
ParameterMRRRaLayer 1: Number of input nodes55Layer 2: Number of membership functions1010Layer 3: Number of rules3232Layer 4: Number of normalization nodes3232Layer 5: Number of type-reduction nodes22Layer 6: Number of output nodes11Footprint of uncertaintyGaussian with uncertain standard deviationGaussian with uncertain standard deviation


### 3.3. Results and Discussion

In the present study, the first 27 data among 30 sets of experimental data are used for training, while the last 3 ones are considered for testing. It should be indicated that process parameters, which were previously discussed in Section3.1, are employed as the inputs of the IT2FNN method and then MRR and Ra are obtained separately as the output of the model. In order to evaluate the performance of the proposed method, the root mean square error (RMSE) scheme is applied to measure the prediction accuracy. The RMSE scheme is defined as follows:(22)RMSE=1N∑k=1Nyk−yak2,where N is the prediction length, while yk and yak denote the kth prediction and actual observation, respectively.Figures3 and 4 show the proposed IT2FNN training process and prediction results for MRR and Ra, respectively. The sample training results (a) show the correlation between the actual and training outputs, while results of the testing sample (b) show the correlation between the actual and the predicted outputs. Moreover, RMSE variation of the training sample during the iterative learning (c) shows the changing trend of the RMSE during the training, while the corresponding prediction errors of the testing sample (d) show the difference between the predicted and the actual results after the training.Figure 3
IT2FNN training process and prediction results for MRR.
(a)(b)(c)(d)Figure 4
IT2FNN training process and prediction results for Ra.
(a)(b)(c)(d)Obtained results show that, after 200 training iterations, the RMSE approached to its asymptotic value and obtained outputs from the model match with the actual measured results during the training so that the prediction accuracy of the test samples increases. Furthermore, it is found that the proposed model has a good generalization ability. Table3 shows the prediction results of MRR and Ra obtained from different schemes, including the ANN [26], ANFIS-GA [27], and ANFIS-MGA [27]. Compared with other models, it is observed that the proposed IT2FNN scheme has the lowest testing RMSE value. Based on the aforementioned results, it is concluded that the proposed method can achieve better prediction performance for MRR and Ra in the WEDT.Table 3
Obtained results from different schemes.
NO.Experimental resultsPrediction result of ANN [26]Prediction result of ANFIS-GA [27]Prediction result of ANFIS-MGA [27]Prediction result of IT2FNNMRRRaMRRRaMRRRaMRRRaMRRRa282.804.2812.463.7342.483.782.6443.8942.77974.2299291.442.7441.322.5991.362.8741.5492.9791.46312.8612300.982.1490.922.5521.031.7910.9252.1380.94512.2544RMSE0.21100.40110.19260.36330.11440.26150.02690.0950


## 3.1. Designing the Experiment

In this section, it is intended to evaluate the performance of the proposed method in predicting the MRR and Ra in the WEDT. The experimental dataset in this paper is published in [26]. The selected material for conducting the experiments is AISI D3 (DIN X210Cr12) tool steel with the Brinell hardness of about 212–248 HB. The diameter of the workpieces and the designed depth of cut are set to 10 mm and 0.1 mm, respectively. Pulse off-time, spark gap, servo feed, rotational speed, and flushing pressure are selected, as shown in Table 1, to analyse their impact on the MRR and Ra in the WEDT.Table 1
Datasets for designing the experiment [26].
No.Pulse off-time (μs)Spark gap (μm)Servo feed (level)Rotational speed (rpm)Flushing pressure (bar)MRR (mm3/min)Ra (μm)130303303.2671.242.396230505703.2672.213.7563308081003.2672.604.1344343051003.2671.733.172534508303.2673.785.009634803703.2671.452.827742308703.2671.502.8998425031003.2670.952.116942805303.2671.683.1331030303301.8931.162.3111130505701.8932.203.53912308081001.8932.474.00313343051001.8931.402.6941434508301.8933.724.8971534803701.8931.372.5021642308701.8931.462.89517425031001.8930.822.0691842805301.8931.653.1241930303301.2631.102.2642030505701.2631.482.86021308081001.2632.223.79422343051001.2631.342.5062334508301.2633.204.5762434803701.2631.352.5202542308701.2631.372.58826425031001.2630.782.0482742805301.2631.542.9872830508703.2672.804.2812934805501.8931.442.7443042503501.2630.982.149


## 3.2. Designing Factors of the Model Architecture

MRR and Ra are separately predicted through the proposed IT2FNN scheme. In this regard, the IT2FNN architecture factors are presented in Table2. The sample data should be initially normalized prior to the training and testing to remove the effect of different dimension scales on the results. The normalization process is defined as follows:(21)x¯i=xi−xminxmax−xmin,where xi and x¯i denote the sampling data and the corresponding normalized data, respectively. Moreover, xmax and xmin are the maximum and minimum values of the sampling data, respectively.Table 2
Design factors in the IT2FNN architecture.
ParameterMRRRaLayer 1: Number of input nodes55Layer 2: Number of membership functions1010Layer 3: Number of rules3232Layer 4: Number of normalization nodes3232Layer 5: Number of type-reduction nodes22Layer 6: Number of output nodes11Footprint of uncertaintyGaussian with uncertain standard deviationGaussian with uncertain standard deviation


## 3.3. Results and Discussion

In the present study, the first 27 data among 30 sets of experimental data are used for training, while the last 3 ones are considered for testing. It should be indicated that process parameters, which were previously discussed in Section3.1, are employed as the inputs of the IT2FNN method and then MRR and Ra are obtained separately as the output of the model. In order to evaluate the performance of the proposed method, the root mean square error (RMSE) scheme is applied to measure the prediction accuracy. The RMSE scheme is defined as follows:(22)RMSE=1N∑k=1Nyk−yak2,where N is the prediction length, while yk and yak denote the kth prediction and actual observation, respectively.Figures3 and 4 show the proposed IT2FNN training process and prediction results for MRR and Ra, respectively. The sample training results (a) show the correlation between the actual and training outputs, while results of the testing sample (b) show the correlation between the actual and the predicted outputs. Moreover, RMSE variation of the training sample during the iterative learning (c) shows the changing trend of the RMSE during the training, while the corresponding prediction errors of the testing sample (d) show the difference between the predicted and the actual results after the training.Figure 3
IT2FNN training process and prediction results for MRR.
(a)(b)(c)(d)Figure 4
IT2FNN training process and prediction results for Ra.
(a)(b)(c)(d)Obtained results show that, after 200 training iterations, the RMSE approached to its asymptotic value and obtained outputs from the model match with the actual measured results during the training so that the prediction accuracy of the test samples increases. Furthermore, it is found that the proposed model has a good generalization ability. Table3 shows the prediction results of MRR and Ra obtained from different schemes, including the ANN [26], ANFIS-GA [27], and ANFIS-MGA [27]. Compared with other models, it is observed that the proposed IT2FNN scheme has the lowest testing RMSE value. Based on the aforementioned results, it is concluded that the proposed method can achieve better prediction performance for MRR and Ra in the WEDT.Table 3
Obtained results from different schemes.
NO.Experimental resultsPrediction result of ANN [26]Prediction result of ANFIS-GA [27]Prediction result of ANFIS-MGA [27]Prediction result of IT2FNNMRRRaMRRRaMRRRaMRRRaMRRRa282.804.2812.463.7342.483.782.6443.8942.77974.2299291.442.7441.322.5991.362.8741.5492.9791.46312.8612300.982.1490.922.5521.031.7910.9252.1380.94512.2544RMSE0.21100.40110.19260.36330.11440.26150.02690.0950


## 4. Conclusions

Studies show that predicting the performance of the machining process is of great significance in the intelligent manufacturing. In practical applications, the precision and the efficiency of the predicting model are two key factors. Obtained results in the present study demonstrate that the IT2FNN is a powerful scheme in dealing with uncertain and nonlinear problems, which is beneficial to predict the performance of the machining process. The proposed method has been used to predict the material removal rate and surface roughness in the wire electrical discharge turning process, which improves the efficiency and precision compared with the related research results. In the near future, it is intended to apply this method for virtual metrology of the machining precision in other machining process fields.



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*Source: 5740362-2020-09-24.xml*