Fuzzy neural network approach to optimizing process performance by using multiple responses

Abstract

This research proposes a method for optimizing process performance; the method involves the use of multiple quality characteristics, fuzzy logic, and radial basis function neural networks (RBFNNs). In the method, each quality characteristic is transformed into a signal-to-noise ratio, and all the ratios are then provided as inputs to a fuzzy model to obtain a single comprehensive output measure (COM). The RBFNNs are used to generate a full factorial design. Finally, the average COM values are calculated for different factor levels, where for each factor, the level that maximizes the COM value is identified as the optimal level. Three case studies are presented for illustrating the method, and for all of them, the proposed method affords the largest total anticipated improvements in multiple quality responses compared with previously used methods, including the fuzzy, grey-Taguchi, Taguchi, and principal component analysis methods. The main advantages of the proposed method are its effectiveness in obtaining global optimal factor levels, its applicability and the requirement of less computational effort, and its efficiency in improving performance. In conclusion, the proposed method may enable practitioners optimize process performance by using multiple quality characteristics.

1 Introduction

Over the past few years, the optimization of multiple responses of a product or process has received considerable attention as a potential technique that can help survive today’s intense competition (Çakıroğlu and Acır 2013; Otebolaku and Andrade 2015). The Taguchi method, which has been traditionally widely used, is applicable only for the optimization of a single quality characteristic of a product or process (Li et al. 2008; Dasgupta et al. 2014). Consequently, several methods have been proposed for optimizing process performance by using multiple quality responses; the methods include data envelopment analysis (Al-Refaie et al. 2009), fuzzy regression (Al-Refaie 2013), artificial neural networks (Zăvoianu et al. 2013; Chen 2015; Wu and Chen 2015), fuzzy methods (Bose et al. 2013), the utility method (Sivasakthivel et al. 2014), and goal programming (Al-Refaie and Li 2011; Al-Refaie et al. 2014). Nevertheless, such methods have several limitations, such as relying on nonparametric evaluation, dealing with a limited number of experiments, achieving local optimality, and requiring good mathematical skills. These limitations and weaknesses of existing methods continue to motivate researchers to develop more efficient methods for solving multiresponse problem in robust design.

In the Taguchi method, finding the combination of optimal factor settings that optimizes the process performance is critical. Although several methods have been proposed in the literature, they have limitations (Al-Refaie 2014a, b, c, d, 2015; Al-Refaie et al. 2012), which include (1) providing local optima; for example, fuzzy logic and regression methods, (2) the methods being nonparametric methods such as the utility concept and grey relational analysis, (3) the relationship between process settings and responses is ignored, (4) insufficiency of concurrent improvements in multiple responses, and (5) requirement of mathematical skills such as knowledge of regression techniques. Accordingly, the present research makes an extension to ongoing research by proposing an approach for optimizing process performance by using multiple quality responses, fuzzy logic, and artificial neural networks (ANNs) techniques. The remainder of this paper is presented in the following sequence. “Fuzzy logic and ANN techniques in the optimization procedure” introduces fuzzy logic, ANN techniques, and the optimization procedure. “Illustrations” provides illustrative case studies. “Research results” summarizes the research results. Finally, the conclusions are presented in “Conclusions”.

2 Fuzzy logic and ANN techniques in the optimization procedure

2.1 Fuzzy logic

The fuzzy logic principle is widely used to handle vague and uncertain information. Two common types of fuzzy systems are used: Takagi–Sugeno (T–S) and Mamdani fuzzy systems. Mamdani fuzzy systems are special cases of T–S fuzzy systems, which involve mathematical expressions that contain a linear function (Lilly 2010). The architecture of a fuzzy system is shown in Fig. 1 (Lilly 2010). The functions of a fuzzy system include fuzzification, which depends on the membership function (MF), rule evaluation, defining the MF for the output, setting the fuzzy rules, and defuzzification to transform the fuzzy value into a comprehensible output measure (Sun and Hsueh 2011; de Pontes et al. 2012).

Fig. 1

Fuzzy logic system. a MFs for the two inputs. b fuzzification of the two inputs

In the proposed method, the center of gravity (COG) defuzzification method is used. Suppose two input variables are to be converted into a COM value by using Mamdani’s fuzzy inference method, the method sets two MFs, high and low, for each input. Let the range of input 1 be [A, D] and the range of input 2 be [B, C]. Then, the fuzzification of the input variables is as shown in Fig. 2. Generally, the rules that relate the two inputs with the output are set as follows:

Fig. 2

MFs for and fuzzification of the two inputs

If input 1 is Low and input 2 is Low then the output is Low.

If input 1 is Low and input 2 is High then the output is Normal.

If input 1 is High and input 2 is Low then the output is Normal.

If input 1 is High and input 2 is High then the output is High.

Applying these rules to fuzzy values yields the following results:

$$\begin{aligned} \mu_{\text{low}} ({\text{input }}1)^ {\wedge} \mu_{\text{low}} ({\text{input }}2) &= \mu_{\text{low}} ({\text{output}}) \hfill \\ \mu_{\text{low}} ({\text{input }}\,1) ^{\wedge} \mu_{\text{high}} ({\text{input }}2) &= \mu_{\text{normal}} ({\text{output}}) \hfill \\ \mu_{\text{high}} ({\text{input }}\,1)^{\wedge} \mu_{\text{low}} ({\text{input }}\,2) &= \mu_{\text{normal}} ({\text{output}}) \hfill \\ \mu_{\text{high}} ({\text{input }}1)^{\wedge} \mu_{\text{high}} ({\text{input }}\,2) &= \mu_{\text{high}} ({\text{output}}) \hfill \\ \end{aligned}$$

For two values A and B, as the representation A ^∧ B denotes min(A, B). For example,

μ _low(input 1) ^∧ μ _low(input 2) = min(μ _low(input 1), μ _low(input 2)).

The MFs for the output are then set as shown in Fig. 3, where S and Q are in the range (0, 1). Finally, output defuzzification is performed by using the COG method to compute the COM value, as shown in Fig. 4.

Fig. 3

MFs for the output

Fig. 4

COG method

Several studies have used the fuzzy logic approach for optimizing performance by considering multiple quality characteristics (Azadeh et al. 2011; Sun et al. 2012; Mandic et al. 2014).

2.2 Artificial neural networks

The ANNs are soft computing techniques based on certain aspects of human behavior; they involve the use of a finite number of layers (which serve as the computing elements) with different neurons. The capabilities of the ANNs are stored in inter-unit connections, strengths, or weights, which are all handled and tuned in the learning process (Asiltürk and Çunkaş 2011; Moosavi and Soltani 2013). The type of ANN that is the most widespread consists of input, hidden, and output layers. The input and output layers represent the variables, and the hidden layer represents the relationship between the input and the output variables. The weights of the nodes are random for a given input pattern and are updated to obtain predicted responses that are less susceptible to errors (Almonacid et al. 2011). In many types of ANNs, such as the backward propagation neural network (BPNN), the gradient descent method, the Levenberg–Marquardt algorithm, the Broyden–Fletcher–Goldfarb–Shanno algorithm, or the resilient back propagation algorithm can be applied to adjust the weights used in the approximation. A multilayer perceptron model is mainly based on the BPNN. The drawbacks of the BPNN are overfitting, and immature decisions that result from the use of local minima rather than global minima. Moreover, a BPNN model shows slow convergence. In the BPNN, a fixed number of neurons should be set before data training, and a wide range of inputs can be covered since sigmoid neurons are used in the hidden layer (Xia et al. 2010). By contrast, the radial basis function neural network (RBFNN) can approximate the desired outputs predicted without requiring a mathematical expression of the relationship between the outputs and the inputs; hence, radial basis functions are called model-free estimators. A high convergence speed is also achievable by using a radial basis function where only one hidden layer is present. The RBFNN architecture is shown in Fig. 5 (Peng et al. 2014). The parameters Ud and yi are the inputs and outputs of the RBFNN for the training data and $G_{l}$ represents the Gaussian function in hidden layer l at the center where the number of controllable factors ranges from 1 to D, n is the number of experiments, the number of hidden layers is L, and $w_{\text{lz}}$ is the weight. The problem of local minima is not encountered in the RBFNN, unlike the BPNN. Because of the advantages and powerful features of the RBFNN, it is widely used for nonlinear functional approximation and pattern classification. The RBFNN has other advantages: local and optimal approximations can be obtained; it has a high convergence rate, high precision, and an adaptive structure; and the output is independent of the weight value set initially (Javan et al. 2013). In the RBFNN, the network output vector is a linear combination of the outputs of the basis function, and the neurons in the hidden layer use the radial basis function as an activation function. If a Gaussian function is chosen as the radial basis function, the outputs can have the following form:

Fig. 5

RBFNN architecture

$$\hat{y}_{z} (s + 1) = \sum\nolimits_{{l = 1}}^{L} {\varphi _{l} } w_{{lz}} = \sum\nolimits_{{l = 1}}^{L} {w_{{{\text{lz}}}} } \exp \left( { - \frac{{\left\| {\hat{U} - m_{l} } \right\|^{2} }}{{2\sigma _{l}^{2} }}} \right)\quad {\text{for}} \, {\mkern 1mu} z = {\mkern 1mu} 1,{\mkern 1mu} 2,{\mkern 1mu} \ldots ,Z$$

(1)

where $\hat{U}(s) = \left[ {\mathop {\hat{U}}\limits_{1} (s) \ldots \ldots \mathop {\hat{U}}\limits_{m} (s)} \right]^{T}$ is the input vector, $\hat{y}_{z} (s + 1)$ is the zth output, l represents the hidden neuron, and the zth output neuron is linked by the weight $w_{lz}$. Furthermore, L denotes the number of Gaussian functions, which is equal to the number of hidden layer nodes, $\varphi_{l}$ represents the Gaussian function in hidden layer l at the center, m _l is the width of the Gaussian function, and $\sigma_{l}$ represents the standard deviation of u _l . In the training algorithm applied to the initial RBFNN, many iterations are performed to find the optimal values of $w_{\text{lz}}$ and $m_{l}$. Therefore, it is imperative to know the initial structure of the network. The input vectors are used for determining the basic functions, and then both input and output data are used to set the weights connected to the output layer. This two-stage training methodology prevents the problem of local minima, and the RBFNN has a high divergence rate (Tsai 2014).

RBFNNs have been widely used in many business applications (Tatar et al. 2013; Chen et al. 2013). Furthermore, several studies have employed the ANN approach for optimizing performance by using multiple quality characteristics. Furtuna et al. (2011) used the ANN method along with a genetic algorithm to optimize the polysiloxane synthesis process. Lin et al. (2012) conducted parameter optimization of a solar energy selective absorption film continuous sputtering process by using Taguchi methods, neural networks, a desirability function, and genetic algorithms. Marvuglia et al. (2014) proposed a new approach involving the combination of a fuzzy logic controller and ANNs for the dynamic and automatic regulation of the indoor temperature.

2.3 The optimization procedure

In the Taguchi method, the columns of the orthogonal array (OA) represent the controllable factors to be studied, and the rows represent combinations of factor levels used in experiments. Typically, there are three types of quality characteristics: smaller-the-better (STB), larger-the-better (LTB), and nominal-the-best (NTB) responses. The proposed approach for optimizing process performance by using multiple quality characteristics is illustrated in Fig. 6 and is outlined as follows:

Fig. 6

The proposed methodology

Step 1: Assume that in an OA, L controllable factors are studied by conducting n experiments to improve J quality characteristics, as shown in Table 1; y _ijk is the kth replicate of the jth response in the ith experiment, where i = 1, …, n; j = 1, …, J; and k = 1, …, K. Compute the signal-to-noise ratio (SNR) ($\eta_{ij}$) for the jth response in experiment i by using the appropriate formula as follows. For STB-type quality characteristic, $\eta_{i}$ is calculated as (Al-Refaie 2010, 2011):

Table 1 Experiment layout

$$\eta_{i} = - 10\log \left[ {1/K\left( {\sum\limits_{k = 1}^{K} {y_{ik}^{2} } } \right)} \right]\quad \forall i$$

(2)

For NTB-type quality characteristic, $\eta_{i}$ is calculated as follows:

$$\eta_{i} = - 10\log \left[ {\mathop {\overline{{y^{2}}} }\limits^{{}}_{i} /s_{i}^{2} } \right]\quad \forall i$$

(3)

where ${\overline{{y_{i}}}}$ and $s_{i}^{{}}$ are the estimated average and standard deviation of response j in experiment i, respectively. For LTB-type quality characteristic, $\eta_{i}$ is calculated as

$$\eta_{i} = - 10\log \left[ {\left( {1/K} \right)\sum\limits_{k = 1}^{K} {1/y_{ik}^{2} } } \right]\;\quad \forall i$$

(4)

The parameter $\eta_{i}$ is computed for each response in experiment i, as shown in Table 2.

Table 2 Signal-to-noise ratios

Step 2: Convert the multiple quality characteristics into a single response by using fuzzy logic (or Mamdani-style fuzzy inference) in which the inputs and output MFs are linear. The inputs are the $\eta_{ij}$ values, whereas the outputs are the COM _i values. The Mamdani-style fuzzy inference process is performed in four stages:

• (1) Fuzzification of the inputs

  Define the MF for each quality characteristic using the corresponding $\eta_{ij}$ values. Let the values G _i1, G _i2, …, G _iJ represent the fuzzy subsets defined by the MFs $\mu_{{G_{i1} }}$, $\mu_{{G_{i1} }}$, …, and $\mu_{{G_{ij} }}$. Use the minimum and maximum values of $\eta_{ij}$ to generate the MF for each quality characteristic, as shown in Fig. 7.

  Fig. 7

  

  MFs for the input variables

• (2) Rule evaluation

  Generate the fuzzy rules that relate the inputs to the output. The fuzzy rule base consists of a set of J inputs, one output measure F, and T rules; for example, for the ith experiment, the rules may be formulated as follows:

  Rule 1: If $\eta_{i1}$ is G ₁₁ and $\eta_{i2}$ is G ₁₂… and $\eta_{iJ}$ is G _1J then F ₁ is M ₁ else

  Rule 2: If $\eta_{i1}$ is G ₂₁ and $\eta_{i2}$ is G ₂₂…. and $\eta_{iJ}$ is G _2J then F ₂ is M ₂ else

  $$\begin{array}{cccccccc} \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ \end{array}$$

  Rule T: If $\eta_{i1}$ is G _T1 and $\eta_{i2}$ is G _T2 …and $\eta_{iJ}$ is G _TJ then F _T is M _T .

• (3) Aggregation of the rule outputs

  From the fuzzy rules, the MFs of the output are identified. For example, the value of M _t that is obtained from the fuzzy rules represents the fuzzy subsets defined by MFs, $\mu_{{M_{t} }}$. To compute M _t for each rule t, the $\eta_{ij}$ value for each quality characteristic is used as an input variable of the rules. The fuzzy reasoning of the rules yields the output by using the max–min composition operation. The MFs of the output of fuzzy reasoning can be expressed as

  $$\mu_{{C_{0} }} (F) = \left( {\mu_{{G_{11} }} (\eta_{i1} ) \wedge \mu_{{G_{12} }} (\eta_{i2} ) \wedge \ldots \ldots \wedge \mu_{{G_{1J} }} (\eta_{iJ} )\mu_{{M_{1} }} (F_{1} )} \right) \vee \ldots \ldots \left( {\mu_{{G_{T1} }} (\eta_{i1} ) \wedge \mu_{{G_{T2} }} (\eta_{i2} ) \wedge \ldots \ldots \wedge \mu_{{G_{TJ} }} (\eta_{iJ} )\mu_{{M_{T} }} (F_{T} )} \right)$$

  (5)

  where ∧ is the minimum operation used in the AND fuzzy operation and ∨ is the maximum operation used in the OR fuzzy operation. Figure 8 shows the fuzzy value for each quality characteristic in experiment i. Suppose two quality characteristics are studied, with the first rule being low for both quality characteristics. This results in low for the output, which means min (low G _i1 and low G _i2).

  Fig. 8

  

  Fuzzy value for each quality characteristic in experiment i

• (4) Defuzzification

  A defuzzification method is used to convert the fuzzy inference output $\mu_{{C_{0} }}$ into a nonfuzzy value COM _i . The conversion is performed using the COG method. The larger the COM value the better the performance. For each experiment i, the COM _i value is calculated using the following equation, and it is displayed in Fig. 9.

  Fig. 9

  

  Defuzzification using the COG method. a MFs for the MMR, b MFs for the SR and c MFs for the EWR

  $$COM = \frac{{\sum F\mu_{{C_{0} }} (F)}}{{\sum \mu_{{C_{0} }} (F)}}$$

  (6)

Step 3: To obtain the global optimal solution, the full data set of the COM is generated by the RBFNN technique on the basis of the input and output matrices; the OA is used as the input matrix and the COM values are employed as the output matrix. The full data set of the COM value is predicted with less error for the full factorial design. Table 3 shows the layout of the two matrices used as the training data and the predicted data for the full factorial design; n and H are the number of experiments in the OA and the number of full factorial runs, respectively. In this approach, the function type of the RBFNN is newrbe, in which the number of hidden layers is two and the spread constant with a smoothing factor of one. The number of neurons is equal to the number of input vectors or the number of experiments. The first layer represents the nonlinear function (a Gaussian function is used as an activation function) and the second layer represents the linear function.

Table 3 Training and predicted data

Step 4: Calculate the average COM values for each factor level, as shown in Table 4. The level that has the largest average COM value is identified as the optimal level for the factor.

Table 4 COM averages for each factor level

Step 5: Compare the total anticipated improvement in each quality response obtained using the proposed approach with those obtained using previously used approaches. The anticipated improvement in each quality response is calculated by subtracting the sum of average SNRs for the combination of optimal factor levels from that for the combination of initial factor levels. Furthermore, calculate the total anticipated improvement in multiple responses and then compare the results.

3 Illustrations

Three real case studies that were investigated in previous studies are adopted for illustrating the proposed method.

3.1 Optimization of process parameters in electro-erosion

Muthuramalingam and Mohan (2014) optimized the parameters of an electro-erosion process by using three quality characteristics: material removal rate (MRR, mm³/min), electrode wear rate (EWR, mm³/min), and surface roughness (SR, μm). The parameters are shown in Table 5. The experimental results of the L₂₇ array are shown in Table 6. In this case study, the quality characteristics are classified into two categories: the EWR and SR are STB and the MRR is LTB. The proposed method is then employed as follows:

Table 5 Experimental parameters of an electro-erosion process

Table 6 Estimated $\eta_{ij}$ values for Case Sudy 3.1

• Step 1: The $\eta_{ij}$ values are calculated for each of the three quality responses for the 27 experiments by using the appropriate formula from Eqs. (2)–(4). Table 6 lists the obtained results.

• Step 2: The Mamdani-style fuzzy inference is adopted to convert the three quality characteristics into a single response. First, the $\eta_{ij}$ values of the MRR, EWR, and SR were fuzzified. The two fuzzy subsets low and high are assigned to the $\eta_{ij}$ values of the MRR, EWR, and SR. Table 7 displays the high and low representations for $\eta_{i2}$ of the SR response as an example. Figure 10 displays the MFs for the three quality responses. The rules that communicate between the MFs of the responses and the output are shown in Table 8. Next, the rule outputs are aggregated, and the four fuzzy subsets are assigned to the output COM value, as shown in Fig. 11. Finally, the fuzzy value of the output is defuzzified to convert it to a crisp COM value for each experiment using the COG method. Table 9 shows the calculated COM _i value for each experiment.

  Table 7 High and low representations of $\eta_{i2}$ for the SR response (μm)

  Fig. 10

  

  MFs for each quality response of Case study 3.1

  Table 8 Generated fuzzy rules for case study 3.1

  Fig. 11

  

  MFs for the COM (Case study 3.1)

  Table 9 Calculated COM values for Case study 3.1

• Step 3: The RBFNN technique shown in Fig. 12 is used to generate the full data set for the COM values. Table 10 displays the full data set obtained.

  Fig. 12

  

  RBFNN architecture for Case study 3.1

  Table 10 Full data set for the COM value for Case study 3.1

• Step 4: The average COM values are calculated for each factor level, as shown in Table 11. It is found that the combination of optimal factor levels is V₁C₁DF₁T₃, whereas it is identified as V₃C₃DF₂T₂ using the grey-Taguchi approach.

  Table 11 Average COM values for the full factorial design for Case study 3.1

• Step 5: For each quality characteristics, the anticipated improvements obtained using the neural–fuzzy approach are compared with those determined using the grey-Taguchi concept.

3.2 Optimization of binary mixture removal

Zolgharnein et al. (2014) employed the Taguchi method and principal component analysis for the optimization of the removal of a binary mixture of alizarin red and alizarin yellow by using multiple LTB quality characteristics: the sorbent capacity of alizarin red (Qr), sorbent capacity of alizarin yellow (Qy), removal percentage of alizarin red (Rr), and removal percentage of alizarin yellow (Ry). The L₂₇ array was used to investigate the process parameters shown in Table 12. The $\eta_{ij}$ values are calculated and displayed in Table 13.

Table 12 Controllable factors and their levels for Case study 3.2

Table 13 Values of $\eta_{ij}$ for the quality characteristics of Case study 3.2

The Mamdani-style fuzzy inference process is performed, in which the two fuzzy subsets low and high are assigned to the four $\eta_{ij}$ values for each quality characteristic (Qr, Qy, Rr, and Ry). The MFs for the COM values are displayed in Fig. 13.

Fig. 13

MFs for the COM for Case study 3.2

To convert the fuzzy value of the output to a nonfuzzy value (the COM value), the COG defuzzification method is used. Finally, the RBFNN shown in Fig. 14 is used to complete the full data set for the COM value, shown in Table 14.

Fig. 14

RBFNN architecture for Case study 3.2. a SR, b TL, c CF and d PC

Table 14 Full data set for the COM value for Case study 3.2

The average COM value for each factor level is calculated and displayed in Table 15. The combination of optimal factor levels is identified as M₂Cr₂Cy₂Ph₁S₂Ti₂T₂ by selecting the levels having the maximum COM value for each factor.

Table 15 Average COM values for the full factorial design for Case study 3.2

The anticipated improvement in each of the quality characteristics (Qr, Qy, Rr, and Ry) is calculated using the neural–fuzzy approach and compared with that obtained using the Taguchi method and principal component analysis.

3.3 Optimization of high-speed CNC turning parameters

Gupta et al. (2011) performed the optimization of high-speed CNC machining of AISI P-20 tool steel by using four LTB-type quality characteristics—SR (µm), tool life (TL, min), cutting force (CF, N), and power consumption (PC, W)—and the Taguchi-fuzzy approach. The selected process parameters presented in Table 16 are studied using the L₂₇ array. The $\eta_{ij}$ values are calculated for each quality characteristic, and they are shown in Table 17. The two fuzzy subsets low and high are assigned to the four inputs by using the $\eta_{ij}$ values of SR, TL, CF, and PC, as shown in Fig. 15. The two fuzzy subsets low and high are assigned to the four inputs by using the $\eta_{ij}$ values of SR, TL, CF, and PC, as shown in Fig. 15. The five fuzzy subsets are then assigned to the output, COM value, as shown in Fig. 16. The COG method is used to convert the fuzzy value to a nonfuzzy value that is called the COM value.

Table 16 Machining parameters and their levels for Case study 3.3

Table 17 Experimental results for Case study 3.3

Fig. 15

MFs for the inputs of Case study 3.3

Fig. 16

MFs of COM for Case study 3.3

The RBFNN technique is used to generate the full factorial data. The COM averages are calculated for each factor level, as shown in Table 18, where the combination of optimal factor levels is identified as S₂F₁D₁N₂E₃. The anticipated improvement for the optimal combination is obtained using the neural–fuzzy approach, and it is compared with that obtained using the Taguchi-fuzzy approach.

Table 18 COM averages for the full factorial design for Case study 3.3

4 Research results

4.1 Optimization results for electro-erosion

The anticipated improvement for electro-erosion is shown in Table 19, where it is clear that for the initial factor settings (V₁C₁DF₁T₁₎, the values of the sum of $\eta_{ij}$ averages for the MMR (LTB), EWR (STB), and SR (STB) are 49.458, −39.041, and 25.680, respectively. For the grey-Taguchi concept (V₃C₃DF₂T₂), they are 38.452, −24.495, and 19.809 dB, respectively. Finally, the fuzzy-RBFNN approach (V₁C₁DF₁T₃) yields the values of the sum for the MMR, SR, and EWR are 33.396, −17.464, and 62.124 dB, respectively. The anticipated improvements in the MMR, SR, and EWR determined using the grey-Taguchi concept (fuzzy-RBFNN) are −11.006 (−16.062), 14.546 (21.577), and −5.870 (36.444) dB, respectively. It is found that the grey-Taguchi concept provides larger anticipated improvement in the MMR (LTB type) compared with the fuzzy-RBFNN approach. By contrast, the fuzzy-RBFNN approach outperforms the grey-Taguchi concept in improving SR and EWR. Finally, the total anticipated improvements for the fuzzy-RBFNN approach (=13.987 dB) are considerably larger than those obtained (=−7.988 dB) using the grey-Taguchi concept.

Table 19 Comparison between the anticipated improvements for Case study 3.1

4.2 Results for optimization of binary mixture removal

The anticipated improvements in Ry, Rr, Qy, and Qr using the Taguchi method and principal component analysis (fuzzy-RBFNN) are 7.4 (1.6), 6.0 (6.4), −10.7 (4.6), and −12.1 (9.5) dB, respectively, as shown in Table 20. It is clear that the Taguchi method and principal component analysis (M₂Cr₁Cy₁PH1S₁Ti₁T₁) provide a larger anticipated improvement in Ry compared with the fuzzy-RBFNN approach (M₂Cr₂Cy₂Ph₁S₂Ti₂T₂). By contrast, the fuzzy-RBFNN outperforms the Taguchi method and principal component analysis in improving Rr, Qy, and Qr. Finally, the total anticipated improvement for the fuzzy-RBFNN approach (=5.5 dB) is considerably larger than that (=−2.4 dB) for the Taguchi method and principal component analysis.

Table 20 Comparison of the improvements for Case Study 3.2

4.3 Results for optimization of high-speed CNC turning

The anticipated improvements in the SR, TL, CF, and PC using the Taguchi-fuzzy approach (fuzzy-RBFNN) are 9.51 (12.83), 7.26 (5.39), 2.75 (3.45), and −0.08 (−0.68) dB, respectively, as shown in Table 21. It is clear that the Taguchi-fuzzy approach (S₁F₁D₁N₂E₃) provides larger anticipated improvements in SF and PC compared with the fuzzy-RBFNN approach (S₂F₁D₁N₂E₃). By contrast, the fuzzy-RBFNN approach outperforms the Taguchi-fuzzy approach in improving the SR and CF. Finally, the total anticipated improvement obtained using the fuzzy-RBFNN approach (=5.5 dB) is considerably larger than that (=4.86 dB) obtained using the Taguchi-fuzzy approach.

Table 21 Comparison of the improvements for Case study 3.3

5 Conclusions

This study successfully integrated fuzzy logic and the RBFNN and used multiple quality responses for the optimization of process performance. For each quality characteristic, $\eta_{ij}$ values are calculated for each quality response, and these values are set as the input to the fuzzy model for obtaining a single response COM. The ANN is then used to predict the full data for the fractional design, for which the full factorial array is used as input to predict the COM value. The average COM values are calculated for each factor level and then adopted to identify the combination of optimal factor levels. Illustrative case studies from past studies are provided. The proposed approach shows appreciable improvements compared with previously used techniques, such as the grey-Taguchi concept, Taguchi method and principal component analysis, and Taguchi-fuzzy approach. Moreover, the proposed technique has the following advantages: (1) it provides a global optimal solution, (2) provides the largest overall improvement, and (3) shows flexibility in dealing with different data sets, increasing number of factors, levels, and factor weights. Future work should consider using the loss function instead of the SNR and/or using genetic algorithms instead of the RBFNN.