Nghiên cứu so sánh mô hình hóa tốc độ loại bỏ vật liệu bằng ANFIS và phương pháp đa thức trong quy trình gia công phóng điện

Contents lists available at ScienceDirect

Computers & Industrial Engineering 79 (2015) 27–41

Computers & Industrial Engineering

j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / c a i e

A comparative study on modelling material removal rate by ANFIS and polynomial methods in electrical discharge machining process

⇑

Khalid Al-Ghamdi, Osman Taylan

a r t i c l e

i n f o

a b s t r a c t

King Abdulaziz University, Faculty of Engineering, Department of Industrial Engineering, P.O. Box 80204, Jeddah 21589, Saudi Arabia

Article history: Received 27 April 2013 Received in revised form 26 October 2014 Accepted 29 October 2014 Available online 7 November 2014

Due to the controversy associated with modelling Electrical Discharge Machining (EDM) processes based on physical laws; this task is predominantly accomplished using empirical modelling methods. The modelling studies reported in the literature deal predominantly with quantitative parameters i.e. ones with numerical levels. In fact, modelling categorical parameters has been devoted a scant attention. This study reports the results of an EDM experiment conducted on the Ti–6Al–4V alloy. Its aim was to model the relationship between the Material Removal Rate (MRR) and the parameters of the process, namely, current, pulse on-time and pulse off-time along with a categorical factor (electrode material). The mod- elling process was accomplished using adaptive neuro-fuzzy inference system (ANFIS) and polynomial modelling approaches. In fact, one purpose of this study was to compare the performance of these mod- elling approaches as no study was found contrasting their prediction capability in the literature. Regard- ing the polynomial model, two numerical parameters (current and pulse on-time) were declared signiﬁcant in the ANOVA together with the electrode material and its interaction with pulse on-time. Thus, they were all incorporated in the developed polynomial model. Furthermore, ﬁve ANFIS models with 6, 9, 19, 21 and 51 rules were developed utilizing the ﬁrst order Sugeno fuzzy approach by back- propagation neural networks training algorithm. Of these, the ANFIS model with 21 rules was the best. This model also outperformed the polynomial model remarkably in terms of predicting error, residuals range and the correlation coefﬁcient between the experimental and predicted MRR values. The study sheds light on the powerful learning capability of ANFIS models and its superiority over the conventional polynomial models in terms of modelling complex non-linear machining processes.

1. Introduction

Recent years have witnessed remarkable increase in the demand for the titanium based alloys owing to their high strength-weight ratio, excellent fracture and corrosion resistance and superior mechanical properties at elevated temperature. An extensive utilization of these alloys has been observed in the aerospace industry (Armendia, Garay, Iriarte, & Arrazola, 2010; Moiseev, 2005). Indeed, many of the aero-engine components are produced using these alloys due, in no small part, to their relatively lightweight which helps to reduce the aircraft’s overall weight thereby moderating fuel consumption (Singh & Khamba, 2007). Moreover, titanium alloys are noticeably replacing the traditionally used materials in aerospace industry such as aluminium (Lütjering & Williams, 2007). They are also used increasingly in other indus-

⇑ Corresponding author. Tel.: +966 500031056; fax: +966 26952486.

Keywords: EDM MRR Polynomial model Neuro-fuzzy model Non-conventional machining

trial and commercial applications such as, chemical processing, pollution control, surgical implantation, nuclear waste storage, food processing and petroleum reﬁning. A major hindrance for the wider use of titanium alloys is the difﬁculties associated with machining them conventionally. Being chemically reactive to almost all tool materials, they tend to weld the cutting tools while machining thereby accelerating tool failure (Hartung, Kramer, & Von Turkovich, 1982). The damage to cutting tools is compounded by the fact that the heat generated at the tool-work-piece interface while machining is concentrated at the tool’s cutting edges. This heat cannot be transferred through the work-piece due to the poor thermal conductivity of titanium alloys. Moreover, a very high mechanical stress is accumulated on the immediate vicinity of the cutting edge as a result of high strength and hardness that these alloys retain at elevated temperature poses a further imped- iment to their machine-ability (Che-Haron & Jawaid, 2005; Trent & Wright, 2000). Ezugwu and Wang (1997) asserted that it was due to the limitations associated with machining these alloys that large companies such as Rolls-Royce and General Electric invested large

E-mail addresses: kaaalghamdi@kau.edu.sa (K. Al-Ghamdi), otaylan@kau.edu.sa (O. Taylan).

formed the examined ANN techniques in predicting the perfor- mance measures whereas comparable performance of the two approaches was reported by (Pradhan & Biswas, 2010).

sums of money in developing techniques to deal with them. Ezugwu, Bonney, Da Silva, and Cakir (2007) argued that, despite these limitations, machining will remain the favoured option for manufacturing complex titanium alloys at a competitive cost in the foreseeable future.

Some gaps were noted in the current literature on modelling die-sinking EDM process. First, while some of the studies com- pared the performance of ANN with that of ANFIS, none of them contrasted the prediction capability of polynomial models with that of ANFIS. This paper attempts to ﬁll this void by comparing the performance of a polynomial model with a Neuro-Fuzzy model derived based on different number of rules and fuzzy inference systems. Second, the parameters examined in pub- lished modelling studies were, by and large, quantitative i.e. their levels were numerical. In fact, scant attention has been accorded to the case of correlating qualitative (categorical) parameters with the key performance measures of EDM, partic- ularly when polynomial modelling was adopted. In this study, the relationship between three numerical parameters along with a categorical one and MRR was modelled using a polynomial function as well as a Neuro-Fuzzy system. The implications of incorporating a qualitative factor and the way in which it was dealt with are elaborated on. Third, in the context of machining Ti–6Al–4V, Caydas and Hascalik (2008) appears to be the only study that addressed the modelling of the EDM process of this alloy. The present work extends that of Caydas and Hascalik (2008) in two aspects. First, while Caydas and Hascalik (2008) examined the inﬂuence of three parameters; namely, pulse on- time, pulse off-time and current on the electrode wear and recast thickness, the current research assessed the impact of the same three parameters (at different levels) along with the effect of electrode material on the MRR, which was not examined in the above mentioned study. Second, although the response sur- face method was the only modelling approach adopted in Caydas and Hascalik (2008), this paper utilizes and compares the perfor- mance of two methods; namely, polynomial modelling and ANFIS, bearing in mind that, to date, the latter has not been applied to model the die-sinking EDM of Ti–6Al–4V.

Non-conventional machining processes provide effective alter- natives to the conventional ones in dealing with the technical dif- ﬁculties associated with machining titanium alloys. Of these, Electrical Discharge Machining (EDM) is the most widely known and used process for the manufacture of engineering components (Aspinwall, Soo, Berrisford, & Walder, 2008). In this process, the material removal takes place as a result of the discharge of energy between a tool and a work-piece, which are separated by a small gap ﬁlled with a dielectric ﬂuid. The process involves a succession of discrete discharge pulses initiated using a DC pulse generator for a certain duration followed by a similar period during which deion- isation of the dielectric occurs and the gap is ﬂushed of debris (McGeough, 1988). A very high temperature is generated by the discharge energy at the point of the spark on the work-piece and the material is removed by melting and vaporization. The stability of the process is maintained by a servo-controlled mechanism whereby a constant gap between the tool and the work-piece is sustained whilst impressing the former pre-selected shape (Singh, Maheshwari, & Pandey, 2004). The EDM theory was postu- lated by two Soviet scientists in the middle of 1940s (Ho & Newman, 2003). Its process is very effective in machining materi- als of any hardness given that they conduct electricity (Abbas, Solomon, & Bahari, 2007; Yan, Chung, & Yuan Huang, 2005). A major advantage of the EDM process is its electro-thermal nature which eliminates the need for any direct contact between the elec- trode and work-piece during machining thereby eradicating the mechanical stress, chatter, tool deformation and vibration prob- lems (Tsai & Masuzawa, 2004). A recent review of titanium alloys EDM is reported in (Kumar, Singh, Batish, & Singh, 2012). The scope of the current work is centred on the EDM of the Ti–6Al–4V alloy, which accounts for 50–60% of total titanium alloy production (Hood, Lechner, Aspinwall, & Voice, 2007).

The methods adopted in the current study are ﬁrstly described. A discussion of the experimental design and procedures follows. The obtained results are then presented and discussed. After com- paring the performance of the investigated modelling approaches, the paper culminates with the main conclusions.

2. Methodology

Material Removal Rate (MRR) is one of the key performance measures in EDM. The ability to rigorously understand its mecha- nism and accurately predict its values based on the well-known physical laws are still debatable (Markopoulos, Manolakos, & Vaxevanidis, 2008; Tsai & Wang, 2001a). In fact, the large number of parameters involved and their interdependence together with the stochastic mechanism of the EDM process appear to signiﬁ- cantly impair the ability to establish a scientiﬁcally admissible theory correlating the process variables with the main perfor- mance measures such as the MRR. Consequently, this task is predominantly accomplished by means of empirical or semi- empirical modelling methods. Broadly, two approaches are adopted; namely, the use of conventional statistical models and the application of soft computing algorithm such as ANFIS. For instance, the second order (quadratic) polynomials, conventionally employed in response surface method, were used to relate the EDM process parameters with its key performance measures in the case of machining cobalt-bonded tungsten carbide (Puertas, Luis, & Alvarez, 2004), silicon carbide (Luis, Puertas, & Villa, 2005), Al–4Cu–6Si alloy–10 wt.% SiCP composites (Dhar, Purohit, Saini, Sharma, & Kumar, 2007) and DIN 1.2714 steel (Zarepour, Tehrani, Karimi, & Amini, 2007). Artiﬁcial neural network (ANN) techniques, on the other hand, were used to model and predict the main EDM performance characteristics in the case of machin- ing aluminium and iron (Tsai & Wang, 2001b) and various steel grades (Panda & Bhoi, 2005; Pradhan & Biswas, 2010). The perfor- mance of ANN was compared with that of the ANFIS in two studies. In one of these (Puertas et al., 2004), the ANFIS method outper-

Design of Experiments (DoE) is an effective and efﬁcient approach for exploring, understanding and empirically modelling the cause-and-effect relationship between the engineering pro- cesses’ parameters and their performance measures. An experi- mental investigation involves purposeful changes of the input variables of a process or system to explore and justify the changes that may occur in the output response (Montgomery, 2010). Generally a DOE study is performed in three stages: plan ning, conducting and analysis and interpretation (Antony, 2003). The planning stage involves recognizing the problem or the improve- ment opportunity, stating the objectives, selecting the perfor- mance measure(s) and the measurement system(s), determining the factors that may inﬂuence the chosen performance measure(s), choosing the levels for the factors, ﬁnding an appropriate design to vary the factor levels in accordance with and assigning the factors and interactions to the selected design. Having accomplished these tasks, the next stage is to carry out the experiment as planned. Finally, the obtained results should be analyzed and interpreted. In so doing, two approaches were adopted in this study; namely, the polynomial modelling approach and the ANFIS.

28 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

2.1. Polynomial modelling approach

(Puertas et al., 2004; Luis et al., 2005; Dhar et al., 2007; Zarepour et al., 2007). For ‘m’ factors, a model of this order comprises (m + 1)(m + 2)/2 terms and takes the form that is given in Eq. (2):

m X

ð2Þ

^y ¼ ^b0 þ

^bixi þ

^biix2

^bijxixj

i þ

i¼1

Polynomial modelling is a major phase of the statistical analysis of experimental data which comprises four stages: estimation, hypothesis testing, modelling and identifying the best parameters setting. The estimation involves obtaining a number to quantify the impact of each of the studied parameters and their interactions on the investigated performance measure(s). To accomplish this task in the current study, a measure that was initially proposed by Kelley (1935) and labelled the Percentage Contribution (PC) by Ross (1996) was adopted. This was obtained using Eq. (1).

PC ¼

(cid:3) 100

ð1Þ

SSP (cid:2) ðdfpÞMSE SStotal

where ^b0 is the intercept, ^bi is the coefﬁcient of the linear effect of factor i, ^bii is the coefﬁcient of the quadratic effect of factor i and ^bij is the coefﬁcient of the interaction effect between factors i and j. Having developed an appropriate empirical model for the process under study, the next step is to identify the best combination of the parameters values. This task can be accomplished using such graphical tools as main effects and interactions plots. Alternatively, formal optimization techniques such as linear and non-linear pro- gramming (Taha, 2010) may be employed using the developed empirical model as an objective function to be maximized or mini- mized as appropriate.

2.2. Adaptive neuro-fuzzy inference system (ANFIS) for EDM processes

Fuzzy systems and ANNs are two of the major soft computing approaches. The relationship between neural networks and linguis- tic knowledge based system is bidirectional and broadly discussed in (Ishibuchi, Nii, & Turksen, 1998). Neural network-based systems can be trained by numerical data and fuzzy rules can be extracted. On the other hand, fuzzy rule-based systems can be designed and developed by linguistic knowledge. Normally, the fuzzy logic and neural systems have very contrasting applications. For instance, fuzzy systems are appropriate if sufﬁcient expert knowledge about the process is available, while neural systems are useful if sufﬁcient measurable data of process are available (Taylan & Karagözog˘lu, 2009). Both approaches build nonlinear systems based on bounded continuous variables. However, they are different in the sense that neural systems are treated in a numerical quantitative manner, whereas fuzzy systems are treated in a symbolic qualitative man- ner. In this study, a class of adaptive neuro-fuzzy networks with the ultimate aim of designing a Fuzzy Inference System (FIS) via learning is proposed. The network structure and learning algorithm imply a systematic approach for the FIS design.

where SSp is the parameter’s Sum of Squares, dfp is its degrees of freedom and the MSE is the mean square error of the experiment. The PC gives the percentage each parameter contributes to the total performance measure’s variation, and is a measure of how much the performance could be improved if the parameter was controlled accurately. The quantities required for estimating the PC can be obtained from the Analysis of Variance (ANOVA) which is the main technique for accomplishing the hypothesis testing task (the process through which the statistical signiﬁcance of each of the examined parameter and interaction is tested). The ANOVA involves decomposing the observed total variance into compo- nents due to different sources of variation (factors, interactions and experimental error). Dividing the estimated variance for each parameter by that of error lends a random variable that follows an F-distribution and can be used to test the statistical signiﬁcance of the concerned parameter. However, when the experiment is not replicated, there will be no degrees of freedom to estimate the error variance. In such cases, it is necessary to identify the factors or interactions that can reasonably be pooled to provide an esti- mate of the error variance. A powerful tool for accomplishing this task is the Half Normal probability Plot (HNP) (Daniel, 1959) of which the use is conventionally conﬁned to analyzing unreplicated two-level experiments. Essentially, HNP can be constructed for any design provided that the plotted effect estimates are uncorrelated and have the same variance. For three-level designs, Wu and Hamada (2011) showed how HNP can be used in the case of decomposing the studied factors into single degree of freedom components with equal variance. Whitcomb and Oehlert (2007) proposed an alternative strategy for using the HNP that relaxed the condition of splitting the effects into single degree of freedom elements. In this strategy, the sums of squares are used as alterna- tive measures of the factorial effects which are commonly esti- mated using the contrasts among the averages of response values associated with the levels of the studied factors. Furthermore, an iterative method that bears considerable resemblance with the stepwise variable selection procedure is adopted to estimate a pro- visional error variance. Dividing the sum of squares of a selected effect by this estimate renders a chi-square statistic for which a provisional p-value can be computed and translated in terms of a provisional z-score of a Half Normal distribution.

The factors and interactions that pronounced signiﬁcant in the ANOVA are used to develop an empirical model correlating them with the response under study (Wu & Hamada, 2011).

An adaptive network is a neuro-fuzzy system with an overall input–output behaviour that is determined by the collection of modiﬁable parameters. The structure of an adaptive network is composed of nodes connected by directed links, where each node performs a function on its incoming signals to generate a signal to the output node and each link speciﬁes the direction of signal ﬂow from one node to another (Jang, Sun, & Mizutani, 1997). The network structure encodes fuzzy ‘If-then’ rules’ in which the conse- quent outcome is a function of the input variables. Fuzzy sets are considered to be advantageous in the logical ﬁeld, and in handling higher order operations easily. The basic advantage of the ANFIS is the addressing of fuzzy linguistic parameters and their term sets that must be identiﬁed by the designer. In general, the designer chooses the types of membership functions (MFs). The number of membership functions for each input space might be a drawback for an ANFIS design besides the parametric learning and structure learning problems dealing with the partition on the input–output universes (Taylan & Darrab, 2011). In order to overcome the draw- back of ANFIS for the rule generation, the fuzzy sub-clustering approach is generally used. In this method, rules are generated based on the clustering of input and output data set. The number of rules is usually equal to the number of output clusters regardless of the number of input variables. Each data point is regarded as a potential cluster centre and a measure of likelihood deﬁning the cluster centre based on the density of surrounding data points is calculated. In order to generate an ANFIS structure, a cluster radius

Polynomials are very powerful for empirically modelling the cause-and-effect relationship between the process parameters (X1, X2..Xm) and the performance measure of interest (Yi). Mathe- matically, this relationship is developed using Taylor series expan- sion of terms involving the studied factors and interactions. Generally, a polynomial model is said to be of order ‘N’ if this is the power of one or more of its factor terms or if ‘N’ is the sum of the powers of the factors involved in one of its interaction terms. Polynomials of second order are commonly used in modelling EDM

29 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

error term ei was deﬁned, as the derivative of the error measure Ep with respect to the output of node i and it is given in Eq. (6).

ð6Þ

ei ¼

@þEp @xi

Then the weight update wki for on-line learning is calculated by

Eq. (7).

ð7Þ

¼ (cid:2)g

Dwki ¼ (cid:2)g

¼ (cid:2)geixk

@þEp @xi

@þEp @wki

@xi @wki

must be speciﬁed to indicate the range of inﬂuence of the cluster. In this approach, specifying a small cluster radius will yield many small clusters in the data set, and will produce too many rules. Hence, the number of MFs will be high and the optimality of outcomes will be lower. To improve the performance of a fuzzy system, the parameters in terms of the structure, complexity, type of networks, etc., have to be determined and tuned (Taylan & Darrab, 2011). On the other hand, an ANFIS model is able to pro- duce crisp numerical outcomes. It includes the following steps: (1) deﬁning input and output variables by linguistic statements, (2) deciding on the fuzzy partition of the input and output spaces, (3) choosing the MFs for the input and output linguistic variables, (4) deciding on the fuzzy control rules, (5) designing the inference mechanism and (6) choosing a de-fuzziﬁcation procedure.

where g is the learning rate that affects the convergence speed and stability of the weights during learning. For off-line learning, the connection weight wki is updated only after the presentation of entire data set or after the speciﬁed epochs are completed as given in Eq. (8).

2.2.1. ANFIS model development by BP MLP

ð8Þ

Dwij ¼ (cid:2)g

; Dwki ¼ (cid:2)g

@þEp @wij

@þEp @wki

This corresponds to a way of using the true gradient direction

based on the entire data set.

The aim of this study is to model the relationship between the MRR and the input parameters of the examined EDM process, namely, current, pulse on-time, pulse off-time and electrode mate- rial. As the modelling task will be accomplished using adaptive neuro-fuzzy inference system (ANFIS) and polynomial modelling, another purpose of the study is to compare the performance of these modelling approaches taking in consideration the presence of a qualitative factor; namely, the electrode material.

3. Experimental design and procedures

The Back-Propagation (BP) Multilayer Perceptrons (MLP) is an adaptive network whose nodes perform the same function on incoming signals; this node function is usually a composite of weighted sum and it is a differentiable nonlinear activation func- tion. The most commonly used activation functions in BP MLPs are logistic, hyperbolic tangent, and identity functions. The ﬁrst two functions provide smooth, nonzero derivatives with respect to input signals. Sometimes they are referred to as squashing func- tions since the inputs to these functions are squashed to the range [0,1] or [(cid:2)1,1]. On the other hand, they are also called sigmoidal functions because of their S-shaped curves that exhibit smoothness and asymptotic properties. For an ANN, the node function for the output layer is considered to be a weighted sum of squashing functions. This is equivalent to a situation in which the activation function is an identity function, and output nodes of this type are often called linear nodes. The net input x of a node is deﬁned as the weighted sum of the incoming signals plus a bias term. For instance, the net input and output of node j are calculated by Eq. (3). and Eq. (4), respectively.

36 X

ð3Þ

Net xj ¼

wijxi þ wj

i¼1

ð4Þ

xj ¼ f ðxjÞ ¼

1 1 þ expð(cid:2)xjÞ

In performing the experiment, an ONA D-2030-S Die sinking EDM machine with programmable pulse current and on/off time was used. Kerosene was used as dielectric and was supplied to the electrode/work-piece interface using two adjustable nozzles at a ﬂow rate of 2 litres/minute. As already pointed out, the work-piece material was the Ti–6Al–4V alloy of which detailed properties are presented in (Ezugwu & Wang, 1997). This was sup- plied in bars which were cut into 15 mm long cylindrical discs each with a diameter of 20 mm. Regarding the electrode, each material type was also provided in bars and was cut into cylindrical discs each with 50 mm length of 12 mm diameter. For each experiment, new electrode was used so that the wear associated with the pre- ceding run exerts no effect on the following ones. This also helped ensure that all the experiments start at the same electrode conditions.

where xi is the output of node i located in any one of the previous layers, wij is the weight associated with the link connecting nodes i and j, and wj is the bias of node j. Since the weights wij are actually internal parameters associated with each node j, changing the weights of a node will alter the behaviour of the node and in turn alter the behaviour of whole BP algorithm. In this study, a BP algorithm forming the steepest descent method for MF estimation was employed.

The training error is the difference between the training data output value, and the output of the fuzzy inference system corresponding to the input training data value. The training error records the mean squared error (MSE) of the training data set at each epoch. The backward error propagation is calculated by the following steps; ﬁrst, a squared error measure for the pth input– output pair is deﬁned by Eq. (5).

ð5Þ

Ep ¼

ðdk (cid:2) xkÞ2

For the EDM process of Ti–6Al–4V, Caydas and Hascalik (2008) conducted an experimental study aiming to develop an empirical model to correlate pulse on-time, pulse off-time and current with electrode wear and recast thickness. A further parameter; namely, electrode material was examined in this study and its inﬂuence together with that of the aforementioned three parameters on the MRR was investigated. Three levels were chosen for each of the studied parameters as detailed in Table 1. A full factorial three-level experiment comprising 34 = 81 runs was conducted. Consequently, there were 80 degrees of freedom to estimate all the investigated factorial effects. Based on the effect of sparsity principle, it was deemed very unlikely that all of these 80 effect components are signiﬁcant i.e. the process is likely to be driven primarily by only some of the parameters’ main effects and interactions. On this premise, the variability associated with the insigniﬁcant effects was attributed to the experimental error and hence was utilized to estimate its variance. By so doing, it was pos- sible to analyze the experimental data without replicating its runs which would have entailed performing a prohibitive number of

where dk is the desired output and xk is the actual output for the node k when the input part of the pth data pair was presented. To minimize the error measure, ﬁrstly the gradient vector was obtained. In order to calculate the gradient vector, a form of deriv- ative beginning from the output layer was identiﬁed which is going backward layer by layer until the input layer is reached. Hence, an

30 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

31 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Table 1 Machining parameters of the Ti–6Al–4V EDM experiment. Variable parameters Levels 1 2 3

irrespective of the pulse on-time, the highest MRR was observed with the graphite electrode. Moreover, for all of the three electrode materials, the MRR increased when the on-time increased from 100 to 150 ls. However, when pulse on-time was changed from 150 to 200, MRR decreased with different rates depending on the electrode material. While a comparable decrease rates were noted with graphite and copper, the least reduction rate was observed with the aluminium electrode. In fact, with long pulse on-time per- iod (>150 ls), arcing was observed with all of the electrode mate- rials. Consequently, the discharge efﬁciency was compromised rendering a noticeable decrease in MRR.

A B C D Pulse current (A) Pulse off-time (ls) Pulse on-time (ls) Electrode material 5 50 100 Graphite 10 75 150 Copper 15 100 200 Aluminium Fixed parameters Level

trials (at least 162). The adoption of principal of sparsity is not uncommon in the experimental design literature (Montgomery, 2010). Its validity has been empirically demonstrated by Li, Sudarsanam, and Frey (2006) and Bergquist, Vanhatalo, and Nordenvaad (2011).

A polynomial model was used to represent the relationship between the parameters and the interaction that were declared signiﬁcant in the ANOVA and the MRR. As a categorical factor, one way of dealing with the electrode material is to develop a sep- arate model consisting only of the quantitative factors for each of its levels rendering three separate models. This approach was adopted by (Sahoo, Routara, & Bandyopadhyay, 2009) in examining the impact of different work-piece materials. Helpful though in terms of avoiding explicit modelling of categorical factors, this approach becomes quite cumbersome as the number of these factors and their levels increases. In fact, explicit modelling of categorical factors has the advantage of being more practical as it yields a single predication equation. Moreover, it renders one esti- mate of the experimental error variance and more residual degrees of freedom than ﬁtting several regression models. Therefore, this approach was adopted in this study.

In each of the performed 81 experiments, the work-piece was machined for a ﬁxed period of time i.e. 15 min using a positive electrode polarity. The order in which the experiments were conducted was randomly determined. The main reason for select- ing a full factorial design factorial design despite its large size was the need to provide enough experimental data to train the devel- oped ANFIS model. The performance measure of interest in the current work was the work-piece MRR. This was measured by recoding the work-piece weight before and after the EDM opera- tion using a micro-level balance; the volume of the material removed was then obtained by dividing the weight differences by the Ti–6Al–4V density. The ratio of the removed material volume to the machining time rendered the MRR (mm3/min).

4. Results and discussion

The results of the experiment are given in Table 2. Their analysis using the two investigated approaches will be presented and dis- cussed separately. The prediction capabilities of the developed polynomial and ANFIS models will then be assessed and compared.

4.1. Polynomial modelling approach

The HNP of the MRR was constructed as shown in Fig. 1. Clearly, pulse current (A), its on-time (C), electrode material (D) and the interaction of pulse on-time and electrode material (CD) stand out as potentially signiﬁcant. To test their statistical signiﬁcance, ANOVA was used as displayed in Table 3. Evidently, all of the examined effects are signiﬁcant because their p-value being smal- ler than a = 0.05.

Electrode material was the most inﬂuential parameter as it accounted for 60.86% of the MRR variability. Pulse on-time had a higher PC (19.43%) than its interaction with electrode material, which explained 9.12% of the variability. Having a PC value of 4.57%, pulse current was the least inﬂuential among the signiﬁcant parameters. Collectively, the considered factorial effects explained 93.98% (60.86 + 19.43 + 9.12 + 4.57) of the MRR variability render- ing an error PC of 6.02% (100–93.98).

To model the process under study, a model matrix was con- structed from the design array given in Table 2. This is a coded matrix that has a column for each term in the model to be devel- oped. As three main effects and one interaction of three-level fac- tors were found signiﬁcant, ten terms were needed. Of these four corresponded to the linear and quadratic components of the two quantitative factors: pulse on-time and current. The qualitative factor (electrode material) has two terms associated with its two degrees of freedom. Finally, the interaction between the electrode material and pulse on-time has four terms corresponding to the four possible combinations of the linear and quadratic elements of pulse on-time and the two components of the electrode mate- rial. To construct the model matrix, it was necessary to code the levels of the studied parameters so that they become dimension- less and can be used to estimate comparable model coefﬁcients. Coding also improves the precision with which the coefﬁcients are estimated (Montgomery, 2010). Linear main effects of the quantitative parameters were coded so that the highest and lowest levels of each factors become 1 and (cid:2)1 respectively. Any value in between can be calculated so that its difference from 1 and (cid:2)1 is proportional to the difference between the corresponding actual value and the actual highest and lowest values. The codes for the quadratic main effect can then be obtained by squaring the coded linear main effect levels. Using this system, a least squares linear main effect coefﬁcient is interpreted as the average change in the response variable per unit change in its factor. The quadratic main effect coefﬁcient, on the other hand, is a measure of how its factor changes when its own values are changed. The components of the electrode material factor can be coded by assigning one dummy variable for each of the three electrode types. For each variable, a value of one can be assigned to the runs where the appropriate electrode material was used while a value of zero should be writ- ten for the remaining runs. This entails adding three dummy vari- able columns of which the row-wise sum is one, rendering a column identical to the one that should be used to estimate the model intercept. Consequently, it will not be possible to estimate any of the model coefﬁcients as the model matrix will be singular. One way out of this problem is to drop one of the three dummy variables, in which case, each of the coefﬁcients of the remaining

Fig. 2(a) depicts the plot of the main effect of current on MRR. Evidently, the MRR increased linearly when the pulse current increased. This is due to the increase in the discharge energy that accompanied the rise of current, which caused an increase in the molten metal volume. With regards to pulse on-time and electrode material, their main effect plots should not be relied upon owing to their interdependence which resulted in subtler patterns of MRR as can be seen from Fig. 2(b). This interaction plot shows that

Dielectric Machining time (min) Dielectric ﬂow rate (l/min) Kerosene 15 2

32 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Table 2 Results of the performed EDM experiment. No. Current Off time On time Electrode No. Current Off time On time Electrode MRR MRR

42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 75 100 100 100 50 50 50 75 75 75 100 100 100 50 50 50 75 75 75 100 100 100 50 50 50 75 75 75 100 100 100 50 50 50 75 75 75 100 100 100 150 150 150 150 200 200 200 200 200 200 200 200 200 100 100 100 100 100 100 100 100 100 150 150 150 150 150 150 150 150 150 200 200 200 200 200 200 200 200 200 Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium Aluminium 15.59 14.75 17.48 19.31 2.25 1.44 8.17 2.03 5.96 4.28 0.91 4.22 5.85 3.75 9 8.35 6.51 8.49 9.04 2.69 4.9 6.37 11.65 19.88 26.04 16.28 11.64 23.49 16.46 12.66 19.06 7.18 14.46 15.96 14.77 13.57 16.89 12.68 17.06 12.04 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 15 5 10 50 50 50 75 75 75 100 100 100 50 50 50 75 75 75 100 100 100 50 50 50 75 75 75 100 100 100 50 50 50 75 75 75 100 100 100 50 50 50 75 75 100 100 100 100 100 100 100 100 100 150 150 150 150 150 150 150 150 150 200 200 200 200 200 200 200 200 200 100 100 100 100 100 100 100 100 100 150 150 150 150 150 Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Graphite Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper Copper 17.21 24.7 23.1 16.45 25.2 31.12 21.39 23.45 30.3 33.69 39.41 36.84 33.73 36.82 38.95 33.31 35.84 38.19 24.75 24.75 30.69 18.69 18.16 27.31 24.14 26.18 27.7 10.27 11.86 17.27 12.5 12.53 16.18 11.68 15.11 11.93 13.9 17.82 19.5 16.76 16.09

Table 3 ANOVA for MRR. Source Sum of squares df Mean square F-value p-value PC (%)

26.56 112.98 353.87 26.50 <0.0001 <0.0001 <0.0001 <0.0001 4.57 19.43 60.86 9.12 179.28 762.68 2388.76 178.91 6.75 A C D CD Residual Total 358.56 1525.35 4777.52 715.66 472.52 7849.61 2 2 2 4 70 80

levels. This system was adopted in this paper for it yields polyno- mial coefﬁcients equivalent to the effect estimates conventionally computed in ANOVA analysis in the cases where the model matrix is orthogonal. To illustrate this, let YG; YC and YA denote respec- tively the average response values associated with the levels graphite, cooper and aluminium of the electrode material and let Y00 be the overall average. For the two components of the electrode material, two main effect coefﬁcients ^bD1 and ^bD2 will be estimated. The former is equal to YG (cid:2) Y00 whereas the latter is equal to YC (cid:2) Y00. Although it is not included in the model, the effect asso- ciated with the third level i.e. aluminium (^bD3) can be obtained from the other two coefﬁcients. In fact, its value equal the sum of two estimated coefﬁcients multiplied by (cid:2)1 i.e. ^bD3 ¼ (cid:2)^bD1þ ^bD2). As it can be seen from the three curves in Fig. 2(b), the MRR has a quadratic behaviour when varied across the values of pulse

two variables will estimate the net contribution or consequence of its corresponding level relative to the excluded one. Alternatively, two columns can be used to represent the two components of the electrode material factor. The ﬁrst can be formed by associating 1, 0 and (cid:2)1 with the levels graphite, cooper and aluminium respec- tively while the second may be constructed by assigning 0, 1 and (cid:2)1 respectively to the graphite, cooper and aluminium electrode

Fig. 1. HNP of the MRR.

33 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

the independence assumption, the violations of the assumption of constant variance can be detected by plotting residuals against ﬁtted values as it is given in Fig. 3 (b).

on-time. This behaviour is dependent upon the electrode material. Put differently, the curvilinear effect of pulse on-time with graph- ite electrode differs from that exhibited with cooper and alumin- ium electrodes. To account for this, the model matrix must incorporate interaction columns. These can be generated using the entry-wise multiplications of the component columns of pulse on-time and electrode material factors. The least square estimates of these interactions serve as adjusting values for the terms in the quadratic equation that represents the behaviour of the MRR over the studied range of pulse on-time for each of the examined elec- trodes. Using the coding scheme described above, the ﬁtted model for MRR is presented by Eq. (9).

MRR ¼ 23:5 þ 2:58XA þ 0:047X2A (cid:2) 0:171XC (cid:2) 9:20X2C

ð9Þ

þ 12:8XD1 (cid:2) 6:72XD2 þ 0:7XCXD1 (cid:2) 4:51XCXD2 (cid:2) 2:93X2CXD1 þ 0:981X2CXD2

As no unusual pattern was recognized in any of the residual plots, the ANOVA results were dammed satisfactory. To assess the model coefﬁcients sensitivity to each of the response values, Cook (1979) suggested a measure, conventionally denoted by Di, which quantify the square difference between the least squares estimates based on all the 81 response values and those obtained by deleting the ith response value. Any response value with Di > 1 is deemed inﬂuential. For the developed model, the maxi- mum Di was 0.129 indicating the robustness of its coefﬁcients to small changes in the response values. Another measure of the model quality is the R2-adjusted which estimates the percentage of the variability explained by the model. This was found to be 93.17% implying that only 6.83% of the total variability in MRR was left unexplained. The predictive capability of the model can also be assessed using the prediction R2 which can be computed as it is given in Eq. (11).

ð11Þ

R2 prediction ¼ 1 (cid:2) ðPRESS=SStotalÞ

ANOVA was used to test the statistical signiﬁcance of the terms that the above model incorporates as it is displayed in Table 4. Clearly, the terms XC, X2A, XCXD1 and X2CXD2 are not signiﬁcant at a = 0.05 level and thus they should be pooled with the error esti- mate and deleted from the developed model to yield the model presented in Eq (10).

MRR ¼ 23:5 þ 2:58XA (cid:2) 9:20X2C þ 12:5XD1 (cid:2) 6:07XD2

ð10Þ

(cid:2) 4:16XCXD2 (cid:2) 2:44X2CXD1

where PRESS is the predictive residual sum of squares (Pe(i)) and e(i) = yi (cid:2) ^yðiÞ. The ^yðiÞ is the predicted value of the ith response var- iable based on a model ﬁt to the remaining n (cid:2) 1 response values. Having an R2 prediction of 92.42%, the ﬁtted model is expected to explain 92.42% of the variability in predicting new response values. Considering the above discussed adequacy measures, the developed model was deemed satisfactory.

4.2. ANFIS modelling approach

In order to validate the conclusions drawn from ANOVA, certain assumptions need to be satisﬁed. These are related to the residuals (the difference between the observed and predicted response values) which should be normally and independently distributed with a mean of zero and constant variance. The normality assump- tion can be assessed by generating a histogram of the residuals as presented in Fig. 3(c) or by plotting them onto a normal probability graph as shown in Fig. 3 (a). While plotting the residuals in time order of data generation (see Fig. 3 (d)) is useful in examining

Fig. 2. Main effect plot of pulse current (a) and interaction of pulse on-time with electrode material (b) based on ANOVA.

In this study, ﬁve ﬁrst order Sugeno ANFIS models were devel- oped based on the learning capability of Back-Propagation Neural Networks (BPNNs) algorithm. The main goal of these ANFIS models is the prediction of MRR in the EDM process. Hence, the available data was divided into training and checking sets. The training data set included 51 observations which served in model building while the checking data set involved 30 observations and served for the validation of the developed model. The developed FIS is able to produce crisp numerical outcomes for the prediction of MRR (mm3/min). The FIS includes choosing the membership functions for the input variables, deciding on the types and number of fuzzy control rules, designing the inference mechanism and choosing a de-fuzziﬁcation procedure.

Table 4 ANOVA for the ﬁtted model. Source Sum of squares df Mean square F-value p-Value

The surface of MRR is plotted in 3D graphs against pulse off time and pulse current (see Fig. 4 (a)), similarly, the MRR was plot- ted against pulse current, and pulse on time (see Fig. 4 (b)), and against pulse off time, and pulse on time (see Fig. 4 (c)).

53.11 0.24 353.87 0.01 225.73 221.06

The nonlinear relations between parameters are clearly appear- ing in these Figs. The 3D plots of parameters are very helpful for viewing the entire output surface of EDM process and the entire

XA XC XD1 X2A X2C XD2 XCXD1 XCXD2 X2CXD1 X2CXD2 358.52 1.59 3285.36 0.04 1523.77 1492.16 87.3 548.55 71.15 8.66 358.52 1.59 2388.76 0.04 1523.77 1492.16 87.3 548.55 71.15 8.66 1 1 1 1 1 1 1 1 1 1 12.933 81.27 10.54 1.28 <0.0001 0.6292 <0.0001 0.9390 <0.0001 <0.0001 0.168 <0.0001 0.001 0.261 6.75 Residual Total 472.52 7849.61 70 80

34 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Fig. 3. Residuals plots for assessing normality (a and c), constant variance (b) and independence (d) assumptions.

the form of

span of the output set MRR based on the entire span of the input set. Fig. 4 (d) depicts two dimensional relations and plots the MRR versus pulse current. Similarly, Fig. 4 (e) and (f) plots pulse on time and pulse off time versus MRR, respectively. In most man- ufacturing problems, relationship between the response and the independent variables is unknown. The 3D and 2D plot of parameters shows that the system is highly nonlinear and suggests the development of intelligent model. The analysis of the MRR surface in relation with the studied parameters implies the existence of many local maximum and minimum responses. This is one of the process complexity indications. In fact, there are many manufacturing processes that cannot be mathematically modelled. Such processes are often satisfactorily controlled by

human beings because of their ability to interpret linguistic state- ments about the process parameters. Fuzzy systems and the other intelligent modelling techniques can provide an effective and real- istic tool to code linguistic statements with imprecise and uncer- tain information (Taylan, 2006). Soft computing techniques are very suitable for modelling the complex manufacturing processes. In these modelling approaches, the domain expert’s intuitions, feelings, and past experiences might be used to judge the magni- tude of the process dynamics and express the system’s parameters to achieve a satisfactory quality level of resulting product. For instance, fuzzy modelling represents the dynamic relations of a process by a set of fuzzy implications with input variables, state and output variables. These implications are fuzzy ‘If-then’ rules

Fig. 4. The 3D (a, b, and c) and 2D (d, e, and f) relations of input parameters versus MRR for ANFIS model.

representing the linguistic relationships of inputs and outputs parameters.

and electrode material and the output parameter of the network is the MRR. For a given input–output data pairs, the BPNN algo- rithm performs two phases of data ﬂow. First, the input parameters pattern was propagated from the input layer to the output layer. The error signals resulting from the difference between the real process outcomes and the network outcomes (MRR) were back propagated from the output layer to the previous layers to update their weights.

The error tolerance was ﬁxed to 0.005 for training the algo- rithm, ﬁve different ANFIS models were constituted with different number of fuzzy rules. The training errors were determined for these models and presented in Table 5. The ANFIS model developed with 21 rules was found to have the minimum percentage of error. The RMSE (Root-Mean-Square-Error) of training error and checking error was found to be 0.3021% and 0.315024% respectively. The developed ANFIS model contained a total of 117 ﬁtting parameters, of which 72 are premise (nonlinear) parameters and 45 are conse- quent (linear) parameters.

4.2.1. Fuzzy rules and membership functions

Fuzzy rules are mathematical relationships mapping the input parameters to output which are constituted from fuzzy linguistic variables and their term sets. Fuzzy ‘If-then’ rules are known as fuzzy implications or fuzzy conditional statements that are wide- spread in our daily linguistic expressions. Fuzzy rules and fuzzy reasoning are the backbone of fuzzy inference system which are the most important modelling tools of fuzzy sets and system (Jang et al., 1997). Fig. 7 illustrates graphical representations of ﬁne tuned membership functions for the input parameters of EDM process together with the linguistic terms set used in this study.

In this study, the number of membership functions were decided by the application of sub-clustering algorithm in which each data point belongs to a cluster at a degree speciﬁed by a mem- bership grade. The idea of clustering algorithm is to divide the out- put data into fuzzy partitions that overlap with input parameters. In this approach, the number of rules has an essential effect on how ﬁne a control level can be achieved on the manufacturing system. Fuzzy rules are constituted from MFs and a FIS is shaped by fuzzy rules. Gaussian membership functions were employed to identify the fuzzy variables in this study. The optimal number of fuzzy rules was considered to be equal to the number of clusters. These fuzzy rules were employed in the inner loop of ANFIS to ﬁne-tune the fuzzy model and obtain the crisp outcomes of MRR. The cluster radius indicates the range of inﬂuence of a cluster when one considers the data space as a unit hypercube. In this approach, specifying a small cluster radius will usually yield too many small clusters in the data which will result in many complications (Jang et al., 1997). Specifying a large cluster radius will usually yield a few large clusters in the data which will result in fewer rules. On the other hand, additional number of MFs does not improve the efﬁciency of the model. The containment of each data to each cluster is deﬁned by a membership function and the degree of membership. The BPNN algorithm was trained based on the exper- imental data of EDM process, by modifying the appropriate param- eters until the outcomes of neural network was close to the actual outcomes. As it is given in Eq. (5), on the basis of gradient descent method for minimization of the error, the correction increments of weighting coefﬁcient were deﬁned to be proportional to the slope related to the changes between the error estimator and the weight- ing coefﬁcients. The weights of algorithm were determined from the input data set. A total of 51 training data and 30 checking data for the EDM process were uniformly sampled from the input ranges and used. The training errors were determined for different number of clusters and the squash factors were presented in Table 5. The error is the difference between the training data outcomes, and the outcomes of the fuzzy inference system corre- sponding to the input parameters. The BPNNs algorithm optimizes the learning procedure by propagating the error to improve the robustness of the fuzzy inference system. Fig. 5 presents the training error of different ANFIS models for different epochs and membership functions, when the ANFIS network accomplished desired training level.

Fig. 8 depicts the fuzzy reasoning procedure for a ﬁrst order Sugeno fuzzy model. Fuzzy reasoning also known as approximate reasoning which is an inference procedure deriving conclusions from a set of fuzzy if-then rules and known facts. Fuzziﬁcation is the initial state of FIS and is deﬁned as mapping from observed inputs to fuzzy sets in a speciﬁed input universe of discourse. A fuzzy set is completely characterized by its MF. A more convenient and concise way of deﬁning a MF is to express it as a mathematical equation. In this study, Gaussian MFs were employed to specify the degrees of membership. Gaussian membership functions are spec- iﬁed by two parameters (c, r), where c represents the MFs center and r determines the MFs width. Fig. 7 plots the Gaussian MFs for the system parameters of ‘pulse current, pulse on time and pulse off time and a singleton MF for the electrode material. If the component ‘pulse on time’ is considered as a fuzzy variable, its membership functions might be {short, medium, and long}. Similarly, the MF of ‘pulse current’ and ‘pulse off time’ are {very low, low, moderate, high, and extremely high} and {short, medium, and long}, respectively. The general illustration of a Gaussian MF is given in Eq. (11). The mathematical equation of MF for the fuzzy linguistic term ‘average’ was constituted as presented in Eq. (12).

Þ2

rð Þ ¼ e(cid:2)1=2 x(cid:2)c

ð11Þ

Gaussian x; c; r

The network under consideration is an ANFIS and has only one overall output. The learning rule speciﬁes how the parameters should be updated to minimize a prescribed errors measure. The rules are mathematical expressions constituting the backbone of FIS and determining the discrepancy between the network’s actual output and desired output. The designed ANFIS Architectures for the EDM process was presented in Fig. 6. It is composed of four nodes in input layer, 21 nodes (H1 (cid:4) H21) in hidden layer and one node (MRR) in the output layer. As it is given in Fig. 6, the inputs of the networks are pulse current, pulse off time, pulse on time

(

for xh100 and xi200

lðpulse on timeÞ ¼ lmoderate ¼

0 e(cid:2)1=2ðx(cid:2)150 25 Þ

for 100 (cid:5) x (cid:5) 200

35 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

ð12Þ

Table 5 The training error of ANFIS model for different epochs.

the membership degree

is 0.726, lpulse

51 1.25 19 1.425 21 1.726 6 1.834 Number of clusters Squash factor Epochs 9 1.525 Error (%)

Similarly, the membership functions of the other fuzzy terms were constituted and the degree of membership was calculated for all the parameters. For instance, when the pulse on time is 150 ls, time (150) = 0.726. When the pulse on time is 180, the membership degree of 180 ls for the fuzzy term ‘moderate’ is 0.487. A similar approach can be used to specify numerically the membership

0 1000 2000 3000 4000 1.2400 0.3924 0.3523 0.3401 0.3124 0.3640 0.3440 0.3356 0.3320 0.3030 1.2500 1.5100 0.9800 0.9000 0.8972 0.3500 0.3034 0.3029 0.3024 0.3021 4.000 3.326 3.100 2.942 2.858

36 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Fig. 5. Plot of training error of ANFIS models with different MFs.

where A and B are fuzzy sets in the antecedents, while zn = fn(x1, x2,...,xm) is a crisp function in the consequent. fn(x1, x2,...,xm) is usu- ally a polynomial in the input variables (x1, x2,...xm), but it can be any function as long as it approximately describes the output of the fuzzy model within the fuzzy region speciﬁed by the antecedent of the rule. Some sample fuzzy rules in the case of multi-input sin- gle output EDM process for MRR (mm3/min) prediction can be pre- sented as follows;

{graphite = 1,

copper = 0,

for

aluminium = (cid:2)1}

degree of each observation by fuzzy linguistic term. As Gaussian MFs were employed in this study, for any speciﬁc observation (xi), a crisp value might be mapped to the fuzzy set with a degree of membership l(xi). Apparently, the universe of discourse of elec- trode materials is discrete and contains non-ordered objects. A fuzzy set whose support is a single point in a universe of discourse with l(xi) = 1 is called singleton. The fuzzy set of electrode materi- als; graphite, copper and aluminium was described with single val- their ues, characterization.

Rule1: IF ‘Pulse current (x1) is low (5A)’ AND ‘Pulse off time (x2) is short (50 lm)’ AND ‘Pulse on time (x3) is long (100 lm)’ AND ‘Electrode Material (x4) is Graphite (1)’ THEN The MRR is 13.5 (mm3/min).

A fuzzy rule set is characterized by the collection of fuzzy ‘If- then’ rules. The collection of fuzzy rules characterizes the simple input–output relation of the manufacturing system. The Sugeno fuzzy model offers a systematic approach to generate fuzzy rules from a given input–output data set. A typical fuzzy rule in a Sugeno model has the following form;

Rule 2: IF ‘Pulse current is moderate (10 A)’ AND ‘Pulse off time is medium (75 lm)’ AND ‘Pulse on time is medium (150 lm))’ AND ‘Electrode Material is Copper (0)’ THEN The MRR (z2 = 0.515x1 - (cid:2) 0.0161x2 (cid:2) 0.0074x3 + 13.2) is 16 (mm3/min).

If x1 is A AND x2 is B; . . . ; Then zn ¼ f nðx1; x2; . . . ; xmÞ

ð13Þ

¼ anx1 þ bnx2 þ cnx3 þ . . . knxm þ rn

Rule 3: IF ‘Pulse current is extremely high (15A)’ AND ‘Pulse off time is short (50 lm)’ AND ‘Pulse on time is long (200 lm)’ AND ‘Electrode Material is Aluminium ((cid:2)1)’ THEN The MRR is 15.9 (mm3/min).

Fig. 6. The architecture of ANFIS model for EDM process MRR.

37 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Fig. 7. The ﬁne tuned MFs for the input parameters of the EDM process.

Fig. 8 shows the fuzzy reasoning procedure for a ﬁrst order Sugeno fuzzy model. For example, to obtain a MRR of 16 (mm3/ min), Rule2 (R #2) was ﬁred. The input parameters of this rule is that the pulse current is equal to 10, pulse off time is equal to 75, the pulse on time is equal to 150 and the electrode material

is copper. The fuzzy reasoning procedure shows that each rule has a crisp output, the overall output (Z) is obtained via weighted average. The weighted average operator is sometimes replaced with the weighted sum operators (zn) presented in Eq. (13). The resulting input–output surface (given in Fig. 4), and the plots of

Fig. 8. Fuzzy reasoning procedures for an EDM process.

the MFs (given in Fig. 7) elucidate the complexity of the modelled EDM process, and show that it could be speciﬁed by the output equation of a fuzzy rule. The fuzzy antecedent part and the linear equation of Rule #2 were presented above. The ﬁrst order outputs of the developed Sugeno fuzzy model are expressed in Eq. (14).

ð14Þ

z2 ¼ 0:515 x1 (cid:2) 0:0161 x2 (cid:2) 0:0074 x3 þ 13:2

z3 ¼ 0:561x1 (cid:2) 0:061x2 þ 12:7

the MRR increases. As per the interdependence between the elec- trode material and the pulse on-time, the pattern that was recog- nized with the data that were generated based on the ANFIS model was similar to the one that was observed in the interaction plot of Fig. 2 (b). However, despite the similarity of the patterns associated with the two models, it is clear that the rates of change that they exhibited are different. Consequently, the predicted val- ues rendered by these models are different and need to be compared.

z4 ¼ 0:561x1 (cid:2) 0:061x2 þ 14:3

4.3. Comparing the performance of developed models

z5 ¼ (cid:2)0:0061x2 (cid:2) 0:0034x3 þ 26:2

z9 ¼ 0:515x1 þ 12:2

z10 ¼ (cid:2)0:0036x3 þ 17:9

z11 ¼ 0:515x1 þ 12:2

z15 ¼ (cid:2)0:0036x3 þ 17:9

In this study, ﬁve ANFIS models together with a polynomial one were developed. The ANFIS models differ in their number of rules which were 6, 9, 19, 21 and 51 rules. Based on the training error, the models that have 9 and 21 rules were the best among the examined ANFIS models. The performance of the developed mod- els was assessed and compared based on the average predictive error and the residual ranges. The average prediction error was computed using Eq. (16).

Average Prediction Error% ¼

PjExperimental value (cid:2) Predicted valuej P Experimental value

(cid:3) 100

ð16Þ

The averages of the prediction error associated with the ANFIS models with 9 and 21 rules were respectively 9.1% and 1.55% whereas the average error of the polynomial model was 10.45%. Clearly, the ANFIS models with 9 and 21 rules outperformed the polynomial model.

Regarding the ﬁring strength of each rule, it is important to note that not all the rules were ﬁred to obtain MRR of 16 (mm3/min) when the inputs of pulse current, its on-time and its off-time were respectively, 10, 150 and 75 and the electrode material was copper. In fact, the ﬁring strength of rule#2 was 1, hence, w2 = 1. The ﬁring strengths of the other rules were as follow: w3 = w4 = w5 = w9 = w10 = w11 = w15 = w17 = w18 = w20 = 0. The weighted sum of linear functions in the consequent is always crisp, the overall crisp output (Z) was determined as the weighted average of each ﬁred rule’s output. Hence, when the pulse current (x1) is equal to 10, pulse off time (x2) is equal to 75, pulse on time (x3) is equal to 150 and the electrode material is copper (rn), the overall outcome is calcu- lated by Eq. (15).

ð15Þ

Z ¼ w1z1 þ w2z2 þ . . . þ wnzn

Z ¼ MRR ¼ 1ð0:515x1 (cid:2) 0:0161x2 (cid:2) 0:0074x3 þ 13:2Þ

þ 0ð0:561x1 (cid:2) 0:061x2 þ 12:7Þ þ . . . . . . þ 0ð(cid:2)0:0036x3 þ 17:9Þ ¼ 16ðmm3=minÞ

The residuals of each of the developed models were obtained by computing the difference between the experimental and the pre- dicted values of MRR as shown in Fig. 10. The residuals ranges from (cid:2)6.5 to 6 for the polynomial models, whereas the ranges were respectively (cid:2)0.73 to 3.71 and (cid:2)9.83 to 6.39 when the ANFIS mod- els with 21 and 9 rules were applied. Being associated with nar- rower residuals range, the polynomial model was marginally better than the ANFIS model with 9 rules; however, the ANFIS model with 21 rules was, by far, the most accurate model for pre- dicting MRR among the examined ones.

The developed ANFIS model can also be used to explain the relationship between the MRR and the process parameters in the same manner that was adopted in interpreting this relationship when the polynomial model was used. To do so, consider the plots of the current main effect and the interaction between pulse on- time and electrode materials which were constructed based on the ANFIS model with 21 rules and depicted in Fig. 9 ((a) and (b)). As was the case in the ANOVA plot, MRR exhibited a linear relationship with the current indicating that as current increases

Linear ﬁts between the experimental data and the predicted MRR values based on the three best models were also developed to assess the degree of correlation between the actual and predicted values as shown in Fig. 11. Noticeably, the ANFIS model with 21 rules outper- formed the other two models. In fact, the coefﬁcient of correlation between experimental MRR data and the predicted MRRs of the polynomial and the ANFIS model with 9 rules was 0.97. However, this value was 0.99 in the case of the ANFIS model with 21 rules s.

38 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Fig. 9. Main effect plot of pulse current (a) and interaction of pulse on-time with electrode material (b) based on ANFIS model.

was smaller than the value obtained based on the polynomial one (9.2%), the superiority of the ANFIS model in approximating the relationship between the MRR and the EDM process parame- ters was conﬁrmed.

39 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Manifestly, the performance of the latter model in predicting the MRR was exceptionally better than the other examined models.

To assess the performance of the examined models in terms of predicting MRR based on new parameter values, a conﬁrmation experiment was performed using new combinations of the factor levels and the results are displayed in Table 6. Also given are the predicted values obtained based on the ANFIS model with 21 rules and the polynomial model together with the error values. These were used to estimate the average prediction error of each model. As the error percentage associated with the ANFIS model (5.07%)

A ﬁnal noteworthy aspect of evaluating the performance of the developed models pertains to their extrapolation capabilities. Besides its superiority in terms of prediction error and the correla- tion between experimental and the predicted MRR values, the developed ANFIS model outperforms the polynomial approxima- tion in the ability to predict MRR values based on factor levels that lie beyond the originally considered range of the quantitative parameters. To elucidate this, it is necessary to reafﬁrm that the experimental data were divided into training and checking sets and that the selection was performed randomly so that no bias affects the allocation of a speciﬁc part of the data to any of the two divisions. Consequently, the checking data were likely to incor- porate input parameter values that lie outside the range of the input values that were included in the training set. Considering that the training algorithm was implemented on the training set and that its performance was assessed based on the error associated with the checking data set which involved new parameter values, it can be concluded that the ability to extrapolate is an essential requirement, based on which, the ANFIS model was developed and evaluated. This is surely not the case with the polynomial mod- elling where the approximation is local and limited to the region of the parameter values used in developing the polynomial function. A relevant warning in this context is that of Rawlings, Pantula, and Dickey (1998) who stated that ‘‘Extrapolations (beyond the X-space

Fig. 10. Residuals of the polynomial and ANFIS with 9 and 12 rules models.

Fig. 11. Linear ﬁts between experimental MRR and its predicted values.

Table 6 The predicted ANFIS MRR outcomes and their errors for EDM process. Error Error Pulse current (A) Electrode material Actual MRR outcomes (mm3/min) ANFIS MRR outcomes (mm3/min) Polynomial MRR outcomes Pulse off time (ls) Pulse on time (ls)

7 12 7 12 7 12 7 12 7 12 7 12 75 75 75 75 75 75 75 75 75 75 75 75 140 140 190 190 140 140 190 190 140 140 190 190 Graphite Graphite Graphite Graphite Copper Copper Copper Copper Aluminium Aluminium Aluminium Aluminium 33.91 35.01 29.63 29.36 16.96 20.16 5.81 7.91 18.25 18.74 18.36 12.19 32.51 36.57 28.95 30.28 15.34 19.42 4.49 7.44 18.10 19.50 21.14 12.10 1.4 (cid:2)1.49 0.73 (cid:2)0.84 1.66 0.76 1.32 0.51 0.15 (cid:2)0.76 (cid:2)2.74 0.09 33.97 36.54 26.99 29.57 16.37 18.95 6.69 9.27 14.49 17.06 14.59 17.16 0.18 4.37 1.21 0.72 (cid:2)3.48 (cid:2)6.00 15.14 17.19 (cid:2)20.62 (cid:2)8.96 (cid:2)20.53 40.77

References

40 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Abbas, M. N., Solomon, D. G., & Bahari, F. M. (2007). A review on current research trends in electrical discharge machining (EDM). International Journal of Machine Tools and Manufacture, 47, 1214–1228. Antony, J. (2003). Design of experiments for engineers and scientists. Butterworth- Heinemann.

Armendia, M., Garay, A., Iriarte, L. M., & Arrazola, P. J. (2010). Comparison of the machinabilities of Ti6Al4V and TIMETAL (R) 54M using uncoated WC–Cotools. Journal of Materials Processing Technology, 210, 197–203.

Aspinwall, D., Soo, S., Berrisford, A., & Walder, G. (2008). Work-piece surface roughness and integrity after WEDM of Ti–6Al–4V and Inconel 718 using minimum damage generator technology. CIRP Annals-Manufacturing Technology, 57, 187–190.

Bergquist, B., Vanhatalo, E., & Nordenvaad, M. L. (2011). A Bayesian analysis of unreplicated two-level factorials using effects sparsity, hierarchy, and heredity. Quality Engineering, 23(2), 152–166.

of the data) are always dangerous but can become disastrous if the equation is not a reasonably correct representation of the true model. Any extrapolation carries with it the assumption that the correlational structure observed in the sample continues outside the sample space. Validation and continual updating are essential for equations that are intended to be used for extrapolation.’’ A fur- ther evidence of the risk associated with extrapolation when using polynomial models relates to the standard deviation (standard error) of any predicted value, which is a quadratic function of the deviation of each input parameter value from its average; and thus increases exponentially as the values of the input parameter move away from their average. Consequently, the prediction of any response value that corresponds to an input parameter level falling beyond the originally considered region is expected to have a signif- icantly larger error than those which lie within the investigated space and have closer values to the input parameters average (Draper & Smith, 1998).

Caydas, U., & Hascalik, A. (2008). Modeling and analysis of electrode wear and white layer thickness in die-sinking EDM process through response surface methodology. The International Journal of Advanced Manufacturing Technology, 38, 1148–1156.

The above discussion revealed the superiority of the developed ANFIS model over the polynomial function and unveiled the level of accuracy and generalizability improvement that can be achieved by adopting this technique in the empirical approximation of the MRR associated with the investigated EDM process.

5. Conclusions

Che-Haron, C., & Jawaid, A. (2005). The effect of machining on surface integrity of titanium alloy Ti–6% Al–4% V. Journal of Materials Processing Technology, 166, 188–192. Cook, R. D. (1979). Inﬂuential observations in linear regression. Journal of the American Statistical Association, 74, 169–174. Daniel, C. (1959). Use of half-normal plots in interpreting factorial two-level experiments. Technometrics, 1, 311–341.

Due to their complexity, the task of modelling EDM processes is predominantly accomplished by means of empirical or semi- empirical modelling methods. Conventional statistical models and soft computing algorithms are broadly the major approaches adopted in the literature. Moreover, most of the reported model- ling studies limited their scope to the examination of quantitative parameters i.e. ones with numerical levels. In fact, modelling cate- gorical parameters has been accorded a scant attention.

International supplies. Dhar, S., Purohit, R., Saini, N., Sharma, A., & Kumar, G. H. (2007). Mathematical modeling of electric discharge machining of cast Al–4Cu–6Si alloy–10wt.% SiCP composites. Journal of Materials Processing Technology, 194, 24–29. Draper, N., & Smith, H. (1998). Applied regression analysis. New York: Wiley. Ezugwu, EO., Bonney, J., Da Silva, R. B., & Cakir, O. (2007). Surface integrity of ﬁnished turned Ti–6Al–4V alloy with PCD tools using conventional and high pressure coolant Journal of Machine Tools and Manufacture, 47, 884–891. Ezugwu, E., & Wang, Z. (1997). Titanium alloys and their machinability – A review. Journal of Materials Processing Technology, 68, 262–274. Hartung, P. D., Kramer, B., & Von Turkovich, B. (1982). Tool wear in titanium machining. CIRP Annals-Manufacturing Technology, 31, 75–80. Ho, K., & Newman, S. (2003). State of the art electrical discharge machining (EDM). International Journal of Machine Tools and Manufacture, 43, 1287–1300.

Hood, R., Lechner, F., Aspinwall, D., & Voice, W. (2007). Creep feed grinding of gamma titanium aluminide and burn resistant titanium alloys using SiC abrasive. International Journal of Machine Tools and Manufacture, 47, 1486–1492. Ishibuchi, H., Nii, M., & Turksen, I. (1998). Bidirectional bridge between neural networks and linguistic knowledge: Linguistic rule extraction and learning from linguistic rules. In Fuzzy systems proceedings, the ieee international conference on computational intelligence (pp. 1112–1117). IEEE.

This study reports the results of an EDM experiment conducted on the Ti–6Al–4V alloy. Its aim was to model the relationship between the MRR and process’s parameters using ANFIS and poly- nomial modelling approaches. Regarding the latter, two numerical parameters (current and pulse on-time) were declared signiﬁcant in the ANOVA together with the electrode material and its interac- tion with pulse on-time. The factors levels were coded so that the linear and quadratic impacts of the numerical factors and their interactions with the two degree of freedom components of the categorical parameters were captured effectively. The R2-adjusted of the developed polynomial model was 0.931 implying that only 6.83% of the total variability in MRR was left unexplained.

Jang, J., Sun, C., & Mizutani, E. (1997). Neuro-Fuzzy and soft computing, New Jersey. Kelley, T. L. (1935). An unbiased correlation ratio measure. Proceedings of the National Academy of Sciences of the United States of America, 21, 554–559. Kumar, S., Singh, R., Batish, A., & Singh, T. (2012). Electric discharge machining of Journal of Machining and International titanium and its alloys: A review. Machinability of Materials, 11, 84–111. Li, X., Sudarsanam, N., & Frey, D. D. (2006). Regularities in data from factorial experiments. Complexity, 11(5), 32–45.

Luis, C., Puertas, I., & Villa, G. (2005). Material removal rate and electrode wear study on the EDM of silicon carbide. Journal of Materials Processing Technology, 164, 889–896.

Lütjering, G., & Williams, J. C. (2007). Titanium. Berlin/Heidelberg: Springer Verlag. Markopoulos, A. P., Manolakos, D. E., & Vaxevanidis, N. M. (2008). Artiﬁcial neural network models for the prediction of surface roughness in electrical discharge machining. Journal of Intelligent Manufacturing, 19, 283–292.

McGeough, J. (1988). Advanced methods of machining. New York: Chapman and Hall. Moiseev, V. (2005). Titanium in Russia. Metal Science and Heat Treatment, 47, 371–376.

Montgomery, D. (2010). Design and analysis of experiments. New York: Wiley. Panda, D. K., & Bhoi, R. K. (2005). Artiﬁcial neural network prediction of material removal rate in electro discharge machining. Materials and Manufacturing Processes, 20, 645–672.

Pradhan, M. K., & Biswas, C. K. (2010). Neuro-fuzzy and neural network-based prediction of various responses in electrical discharge machining of AISI D2 steel. The International Journal of Advanced Manufacturing Technology, 50, 591–610.

Five ANFIS models with 6, 9, 19, 21 and 51 rules were developed. In so doing, ﬁrst order Sugeno fuzzy approach was adopted in con- junction with back-propagation neural networks training algo- rithm. The data were divided into training and checking sets. The former included 51 observations which served in model building while the latter involves 30 observations and was used for the pur- poses of checking and validating the developed models. Of the ﬁve developed ANFIS models, the ones with 9 and 21 rules were the best. The performance of these was compared with that of the poly- nomial model. While the ANFIS model with 9 rules had a compara- ble performance with the polynomial model, the one with 21 rules outperformed both of them based on prediction error, residuals range and the correlation coefﬁcient of the experimental and pre- dicted MRR values. The superiority of the ANFIS model was veriﬁed based on a conﬁrmation experiment that was conducted based on new combinations of the input parameters values. This can be attributed to the ﬂexible learning capability that ANFIS models pos- sess which can also be advantageous for extending the use of such approach from modelling to controlling EDM processes thereby paving the way for interesting future research topics.

Puertas, I., Luis, C., & Alvarez, L. (2004). Analysis of the inﬂuence of EDM parameters on surface quality, MRR and EW of WC–Co. Journal of Materials Processing Technology, 153, 1026–1032. Rawlings, J., Pantula, S., & Dickey, D. (1998). Applied regression analysis: A research tool. New York: Springer.

Ross, P. J. (1996). Taguchi techniques for quality engineering. New York: McGraw-Hill. Sahoo, P., Routara, B. C., & Bandyopadhyay, A. (2009). Roughness modeling and optimization in EDM using response surface method for different work piece materials. International Journal of Machining and Machinability of Materials, 5, 321–346.

41 K. Al-Ghamdi, O. Taylan / Computers & Industrial Engineering 79 (2015) 27–41

Singh, R., & Khamba, J. (2007). Investigation for ultrasonic machining of titanium and its alloys. Journal of Materials Processing Technology, 183, 363–367. Tsai, K. M., & Wang, P. J. (2001a). Comparisons of neural network models on material removal rate in electrical discharge machining. Journal of Materials Processing Technology, 117, 111–124.

Singh, S., Maheshwari, S., & Pandey, P. (2004). Some investigations into the electric discharge machining of hardened tool steel using different electrode materials. Journal of Materials Processing Technology, 149, 272–277. Tsai, K. M., & Wang, P. J. (2001b). Predictions on surface ﬁnish in electrical discharge machining based upon neural network models. International Journal of Machine Tools and Manufacture, 41, 1385–1403.

Whitcomb, P, & Oehlert, G. W. (2007). Graphical selection of effects in general factorials, 2007 fall technical conference of the American Society for Quality (ASQ) and the American Statistical Association (ASA). Jacksonville, FL. Wu, C., & Hamada, M. (2011). Experiments: Planning, analysis, and parameter design Taha, H. (2010). Operations research: An introduction. New York: Prentice Hall. Taylan, O. (2006). Neural and fuzzy model performance evaluation of a dynamic production system. International Journal of Production Research, 44, 1093–1105. Taylan, O., & Darrab, I. A. (2011). Determining optimal quality distribution of latex weight using adaptive neuro-fuzzy modeling and control systems. Computers & Industrial Engineering, 61, 686–696. optimization. New York: Wiley.

Yan, B. H., Chung, Tsai H., & Yuan Huang, F. (2005). The effect in EDM of a dielectric of a urea solution in water on modifying the surface of titanium. International Journal of Machine Tools and Manufacture, 45, 194–200. Taylan, O., & Karagözog˘lu, B. (2009). An adaptive neuro-fuzzy model for prediction of student’s academic performance. Computers & Industrial Engineering, 57, 732–741.

Zarepour, H., Tehrani, A. F., Karimi, D., & Amini, S. (2007). Statistical analysis on electrode wear in EDM of tool steel DIN 1.2714 used in forging dies. Journal of Materials Processing Technology, 187, 711–714. Trent, E. M., & Wright, P. K. (2000). Metal cutting. Boston: Butterworth-Heinemann. Tsai, Y. Y., & Masuzawa, T. (2004). An index to evaluate the wear resistance of the Journal of Materials Processing Technology, 149, electrode in micro-EDM. 304–309.