Modelling and multi-response optimization of wire electric discharge machining parameters using response surface methodology and grey–fuzzy algorithm

Abstract

Wire electric discharge machining is a non-conventional machining wherein the quality and cost of machining are influenced by the process parameters. This investigation focuses on finding the optimal level of process parameters, which is for better surface finish, material removal rate and lower wire consumption for machining stainless steel-316 using the grey–fuzzy algorithm. Grey relational technique is applied to find the grey coefficient of each performance, and fuzzy evaluates the multiple performance characteristics index according to the grey relational coefficient of each response. Response surface methodology and the analysis of variance were used for modelling and analysis of responses to predict and find the influence of machining parameters and their proportion of contribution on the individual and overall responses. The measured values from confirmation experiments were compared with the predicted values, which indicate that the proposed models can be effectively used to predict the responses in the wire electrical discharge machining of AISI stainless steel-316. It is found that servo gap set voltage is the most influential factor for this particular steel followed by pulse off time, pulse on time and wire feed rate.

Introduction

Austenitic stainless steel (SS)-316 is one type of marine grade steel and has various applications in the field of defence, nuclear science, critical structural components in chemical industries, pharmaceutical equipment, photographic developing equipment and medical implants because of its excellent mechanical properties and corrosion resistance in saline environment. The existence of molybdenum (2%–3%) and chromium (12%–25%) in the steel prevents corrosion.^1–3 SS-316 is a very difficult-to-machine material even with specific coated tools because of its high strength, high work-hardening tendency, poor thermal conductivity and abrasive behaviour.^2,4 During the conventional cutting of such steel, cutting tool rides on the surface of the material and in consequence of which heat is produced on the surface. Then, the workpiece surface becomes harder due to rapid cooling even at low deformation rate. Sometimes, unusual stress or residual stress is developed due to high heat concentration at different zones of the machine surface.^2,5 This phenomenon may lead to form micro-cracks and unusual hardness on the surface.

To overcome this problem, wire electric discharge machining (WEDM), which is one of the abrasionless non-conventional machining processes, may be adopted to cut such kind of SS. The main principal of WEDM is erosion of the material from the workpiece in a form of debris by occurrence of series of discrete sparks between the two electrodes separated by a continuous stream of dielectric fluid. A thin copper wire of diameter 0.1–0.3 mm is happened to be consumed in this process. ⁶ However, the main drawback of the process is the relatively low machining speed and high tool cost, as compared to the other cutting processes. Thus, selection of optimum machining parameters in such process is very important to satisfy all the conflicting objectives such as quality and cost of the process.

Some attempts were made to find out machinability characteristics. Muhammad et al. ⁷ studied heat-affected zone (HAZ), kerf width, surface roughness (SR) and dross deposition of such material during fibre laser profile cutting. Somashekhar et al. ⁸ measured the crater size of each spark and temperature distribution on the SS-316L workpiece by performing transient thermal analysis using micro-electric discharge machining (μEDM). Durairaj et al. ⁹ tried to summarize the grey relational theory and Taguchi optimization technique, in order to optimize the cutting parameters in WEDM for SS-304 to achieve the minimum kerf width and the best surface quality individually. Nayak and Mahapatra ¹⁰ had considered the three important output performance measures, namely, angular error (AE), SR and cutting speed (CS), for optimizing machining parameters of WEDM taper cutting process for SS-304. The authors mainly highlighted on a Taguchi-utility optimization approach to determine the optimal process parameters in WEDM process during taper cutting operation. Raju et al. ¹¹ investigated the affects of parameters such as pulse on time, peak current, servo voltage and wire tension (WT) on SR of SS-316 using a two-level L16 orthogonal array in WEDM machine (Japax LDM 50). From analysis of variance (ANOVA), it was found that pulse on time is a significant factor on SR. Pulse on time and current have direct effect on SR, whereas by increasing servo voltage, SR gets reduced. Other parameters such as pulse off time and wire feed (WF) rate also have great influence on SR which was not mentioned in that article. The signal-to-noise (S/N) ratio was only used to evaluate quality characteristics, that is, SR by conducting the experiments. Some other important measures of performance such as material removal rate (MRR) and wire consumption (WC) are still uncovered to fulfil multi-objective problem.

A WEDMed surface quality is mainly characterized by its roughness. ¹² SR is a very important criterion for selection of any machining condition. It is a representation of the technological quality of a product such as the surface textures of the machined parts. ¹³ MRR is a crucial response in determining the productivity of the WEDM process.^14,15 Higher MRR in WEDM is caused due to the lower melting temperature and electrical resistivity of this material. ¹⁶ The tool cost of WEDM is high as compared to other cutting processes. So, it is also important to minimize tool consumption for WEDM.

An efficient prediction method called response surface method has been implemented here for predicting such process responses in various combinations of factor setting. Response surface methodology (RSM) also helps to find interaction effect among the control parameters on responses. ¹⁷ Golshan et al. ¹⁸ presented a mathematical model using RSM method to optimize the process parameters and analyse the model to signify the process behaviour of WEDM operation.

For solving the complex interrelationships between the multiple responses, grey relational analysis theory is used. But it contains some degree of uncertainty for a definite response. ¹⁹ The fuzzy logic theory which is an effective mathematical module of resolving problems contains the uncertain huge information.^20,21 In order to achieve crisp values, three steps, namely, fuzzification of input variable, rule inference and defuzzification, are included in fuzzy logic approach. ²⁰ The defuzzification method adopts the fuzziness of human concepts to deal with multiple responses. ²² Lin et al. ²³ applied fuzzy logic to perform a fuzzy reasoning of the multiple performance characteristics. As a result, these approaches can greatly improve the process responses such as the electrode wear ratio, MRR and SR in the EDM process.^24,25 Lin et al. ²⁶ reported the grey relational analysis and fuzzy-based Taguchi method for optimizing the same responses in EDM process. It has been found that the grey relational analysis is more straightforward than the fuzzy-based Taguchi method for multi-objective optimization of the EDM process. The fuzzy logic coupled with grey relation approach improves the grey relational grade and results in absolutely lesser uncertain output than that of grey relational approach alone. The grey grade output from fuzzy logic system (grey–fuzzy reasoning grade) must be more accurate than that of grey relational grade. ²⁷ The grey–fuzzy logic analysis converts the complex multi-performance characteristics optimization problem into the optimization of an easy single grey–fuzzy reasoning grade. ²⁸ From these literatures, it has been found that grey–fuzzy is an effective tool to correct optimization of multi-objective problem.

The ANOVA is a set of statistical methods and mathematical functions used to identify the significant factors in a multi-significant model. ²⁹ The ANOVA is performed to identify the contribution of the different process parameters on those multiple performance characteristics by decomposition of variance. ³⁰ In this study, it is used to determine the type of significance at 95% confidence interval. The process parameters for which p-value is greater than 0.05 indicate that those are not significant at 95% confidence interval. ³¹

This study focuses on modelling and multi-objective optimization of WEDM for machining SS-316, considering multiple responses such as Ra, MRR and WC. It is pertinent to mention here that prediction of Ra, MRR and WC using RSM method and the optimization of WEDM process parameters on overall performance using grey–fuzzy method are not reported anywhere for better surface quality with minimum manufacturing cost.

Grey relational analysis and fuzzy logic are combined to establish the optimal set of control machining parameters. RSM is applied to predict clearly Ra, MRR and WC, which cannot accurately be predicted by grey–fuzzy method. Finally, ANOVA easily discovers the significant process parameters by analysis of data and checking the adequacy of the model.

Methodology

Modelling

RSM

RSM was first developed by Box and Wilson; ³² this model is generated from the designed experiment and is verified by means of statistical techniques. To modify the extensive full factorial design, a three-level incomplete factorial design called Box–Behnken design (BBD) was developed by Box and Behnken. ³³ BBD and RSM are used to plan the experiment ³⁴ and help in quantifying the relationship between various process parameters with various machining criteria and find the effects of these process parameters on the measured responses. ³⁵

Multi-objective optimization

Grey relational analysis

The grey theory is used to solve uncertain and discrete data which have complicated interrelationship among the multiple performance characteristic problems. Calculation of grey relational analysis involves the following steps:^26,36–38

Step 1. Calculation of S/N ratio to compute quality characteristics.

MRR, Ra and WC are conflict objectives, where MRR is considered as ‘higher the better’ and Ra and WC are considered as ‘lower the better’.

The S/N ratio of MRR is computed as ³⁹

$$\eta =-10{\log }_{10}\lceil \frac{1}{n}\sum _{i=1}^n\frac{1}{y_{\mathrm{ij}}^2}\rceil$$

(1)

The S/N ratio for Ra and WC is computed as

$$\eta =-10{\log }_{10}\lceil \frac{1}{n}\sum _{i=1}^n{y}_{\mathrm{ij}}^2\rceil$$

(2)

where n is the number of tests and y_ij is the response variable, i = 1, 2, …, n; j = 1, 2, …, m (m is the number of response).

Step 2. Normalization of S/N ratio of the quality characteristics.

When the range of the data is too large, that is, one data and its unit differ from other, some factors will be ignored. To eliminate such effect, S/N ratio of the quality characteristics is normalized first.

The normalization of S/N ratio is taken by the following equation

$$X_{\mathrm{ij}}(m)=\frac{{\eta }_i(m)-min{\eta }_i(m)}{max{\eta }_{\mathrm{ij}}-min{\eta }_{\mathrm{ij}}}$$

(3)

where $X_{\mathrm{ij}}(m)$ are the normalized data of m element in the ith sequence; ${\eta }_i(m)$ denotes the corresponding S/N ratio and $max{\eta }_{\mathrm{ij}}$ and $min{\eta }_{\mathrm{ij}}$ are the largest and smallest values of ${\eta }_{\mathrm{ij}}(m)$ , respectively.

Step 3. Calculation of grey relational coefficient.

Grey relational coefficient is the sequences used in grey relational analysis and it is used for determining how close $X_i(m)$ and $X_0(m)$ are. It can be calculated as

$$\xi (m)=\frac{{\Delta }_{\min }+\zeta {\Delta }_{\max }}{{\Delta }_{\mathrm{oi}}(m)+\zeta {\Delta }_{\max }}$$

(4)

i = 1, 2, …, m

$${\Delta }_{\mathrm{oi}}(m)=|X_0(m)-X_i(m)|$$

(5)

where ${\Delta }_{\min }$ is the minimum deviation sequence of ith sequence of mth element =  $min\{{\Delta }_i(m)\}$ ; ${\Delta }_{\max }$ is the maximum deviation sequence of ith sequence of mth element =  $max\{{\Delta }_i(m)\}$ and $\zeta$ is the distinguishing coefficient and its value is in between 0 and 1. The smaller $\zeta$ implies higher distinguishability. (In general, it is set to $\zeta =0.5$ .)

Step 4. Multiple performance characteristics index (MPCI).

Multi-response problems are converted into a single-objective optimization (called MPCI) by feeding grey relational coefficient value of each response into fuzzy system. The weighted sum of the grey relational coefficients is counted as the MPCI. Here, the weightage of individual responses is generated using fuzzy rules to calculate MPCI. The highest value of MPCI evaluated the optimal parametric setting.

Fuzzy logic

The grey relational coefficient ξ_i (m) for each sequence contains a certain degree of uncertainty and vagueness in the definition of response. In fuzzy logic analysis, initially, the fuzzifier uses membership functions to fuzzify the grey relational coefficient. The inference engine performs a fuzzy reasoning on fuzzy rules to generate a fuzzy value. Finally, the defuzzifier converts the fuzzy value into an MPCI. A general fuzzy inference system has basically four components: fuzzification, fuzzy rules base, fuzzy output engine and defuzzification.^24,26,36

Fuzzification

Fuzzification converts each grey relational coefficient into degrees of membership. A membership function is a curve that defines how each input value is mapped to a membership value between 0 and 1. Triangular membership functions are used here because of their simplicity and are easy to implement in a computer programme.

Fuzzy rule base

After fuzzification of input process parameters and linguistic representation of output variable, fuzzy rules established a fuzzy relation between the input attributes and output goals. ‘IF–THEN’ format is used in expressing these rules. In this study, the Mamdani-type fuzzy rules are constituted.

Fuzzy inference engine

The fuzzify inference engine will perform a fuzzy interface on fuzzy rules in order to generate a fuzzy value. In the fuzzy logic, if–then rule statements are used to formulate two grey relational coefficients ξ_n and one multi-response output y, that is

Rule. If ξ₁ is A_i1 and ξ₂ is A_i2, then y is D_i

where A_i, B_i and D_i are the fuzzy subsets defined by the corresponding membership functions, that is, μ_Ai, μ_Bi and μ_Di. The fuzzy output y is provided from those above rules by employing the max–min interface operation. The inference results in a fuzzy set with membership function for the MPCI y can be expressed as follows

$${\mu }_{D0}(y)=\left(\begin{array}{c}\hfill {\mu }_{A1}({\xi }_1)\wedge {\mu }_{B1}({\xi }_2)\wedge {\mu }_{D1}(y)\vee \dots \hfill \\ \multicolumn{1}{c}{{\mu }_{\mathrm{An}}({\xi }_1)\wedge {\mu }_{\mathrm{Bn}}({\xi }_2)\wedge {\mu }_{\mathrm{Dn}}(y)}\end{array}\right)$$

(6)

where ∧ and ∨ are the minimum and maximum operations, respectively.

Defuzzification

The final step is defuzzification; it is the process of converting a fuzzy set into a non-fuzzy value which will be called as the MPCI in this study. The fuzzy multi-response output μ_D0(y) must be transferred to a non-fuzzy value y₀ by the calculation of centroid defuzzification method, that is

$$y_0=\frac{\sum y{\mu }_{D_0}(y)}{\sum {\mu }_{D_0}(y)}$$

(7)

This non-fuzzy value y₀ is the so-called MPCI. It has been observed that the higher MPCI value implies the better product quality. MPCI values are converted into the corresponding S/N ratio using ‘larger the better’ criteria.

ANOVA for identifying the significant factors

The main purpose of the ANOVA is to identify the affect of each factor on individual responses and overall responses. The results of ANOVA will determine very clearly the impact of each factor on the MPCI. Usually, the larger ‘F’ value indicating the change in the process parameter has a significant effect on the performance characteristics.

Experimental details

Procedure

During this study, a series of experiments were conducted using ELEKTRA SPRINTCUT CNC wire electrical discharge machine. Brass wire was used as tool electrode. During all the experiments, the workpiece was submerged in deionized water (dielectric fluid). Austenite standard chromium nickel steel AISI 316 was used to machine using WEDM. The specification of the work material is shown below:

AISI SS-316: (%) C 0.03, N 0.1–0.3, P 0.045, Ni 10–14, Cr 16–18, Mn 2.0, Si 1.0 and Mo 2–3.

The SR, MRR and WC had been measured for each experiment. Each combination of the experiment is conducted three times and one of them is reported in Table 2. But all three replicated values are used for analysis.

Performance characteristic evaluation

Average SR (Ra)

The WEDM process parameters have great influence on the surface integrity like Ra. Ra is the arithmetic mean of the absolute departures of the roughness profile from the mean centre line along the sampling length. ¹³ The average roughness of the surface after WEDM was measured by a three-dimensional (3D) surface analyser (Talysurf CCI Lite). The results are also cross checked by portable style–type profilometer, Talysurf (Model: Taylor Hobson) with 0.8 mm cut-off length and 15 mm sample length. The unit of SR is in micrometer.

MRR

MRR has been used to evaluate machining performance. Material removal was calculated from the difference in weight of the workpiece before and after the experiment. MRR can be measured as follows

$$\mathrm{MRR}=\frac{({\mathrm{W}}_{\mathrm{i}}-{\mathrm{W}}_{\mathrm{f}})}{{\rho }_{\mathrm{s}}\mathrm{t}}{\mathrm{mm}}^3/\min$$

(8)

where W_i is the initial weight of the workpiece in gram, W_f is the final weight of the workpiece, t is the machining time in minutes and ρ_s is the density of the steel (7.8 × 10⁻⁶ kg/mm³).

WC

The wire eventually eroded or broke down after operation which cannot be reused or recycled. The wire after being used once has a different diameter and tensile strength than it was started with. The reason is that it has lost its coating (if it was a coated wire) and tensile strength. Then, it is nothing other than good for scrap. So the consumption of the wire is one of the most important points in case of cost analysis. It can be measured from the difference in weight of spool before and after the operation. The unit of WC is in kilogram.

Experimental design

The process variables investigated in this experiment included pulse on time, pulse off time, pulse current, WF, WT and spark gap set voltage. The parameters and their units and range are selected for conducting the experiments which have been reported in Table 1.

Table 1.

Machining parameters.

Process parameters	Machine unit (mu)	Level (in)
		1	2	3
Ton	mu (μs)	110 (0.6)	115 (0.85)	120 (1.1)
Toff	mu (μs)	50 (26)	55 (36)	60 (46)
Ip	A	210	220	230
WF	m/min	4	6	8
WT	mu (kg)	4 (0.5)	6 (0.7)	8 (1)
SV	V	15	20	25

WF: wire feed; WT: wire tension.

The equal-energy pulse (e-type pulse) has been chosen because of homogeneity of spark and energy for every spark. The ranges of parameters have been selected with the help of trial experiment and past work. Beyond this range, that is, at higher values of T_on, Ip and WT and lower values of T_off and SV, wire breaks rapidly, and thus a very rough surface is obtained. The lower values of T_on, Ip and WT and higher values of T_off and SV cause very low MRR and huge WC. RSM L54 BBD of the experiment was employed to conduct the experiments. The experimental design and the results are shown in Table 2.

Table 2.

Design of experiments of experimental data.

S. no.	Ton	Toff	Ip	WF	WT	SV	Ra (μm)	MRR (mm3/min)	WC (kg)
1	110	55	220	4	8	20	2.98	2.9210	0.032319
2	115	55	220	6	6	20	3.26	3.5164	0.027308
3	120	60	220	8	6	20	3.95	3.8864	0.029186
4	115	60	230	6	4	20	3.45	2.7619	0.036198
5	120	55	220	8	8	20	3.81	4.4885	0.023573
6	120	55	230	6	6	15	4.05	4.4932	0.019866
7	115	60	220	6	8	15	3.68	2.8740	0.032674
8	115	50	210	6	4	20	3.01	3.5012	0.02632
9	115	55	220	6	6	20	3.32	3.5060	0.029438
10	115	55	230	4	6	15	3.50	3.6562	0.019367
11	115	55	230	4	6	25	3.28	3.4491	0.022997
12	110	55	220	8	8	20	2.85	2.3934	0.047071
13	120	60	220	4	6	20	3.82	4.0312	0.013681
14	120	55	220	4	8	20	3.56	4.8647	0.012721
15	110	60	220	4	6	20	2.79	1.6708	0.0421
16	115	55	210	4	6	25	2.60	3.2676	0.024031
17	110	55	220	4	4	20	2.75	2.1066	0.03583
18	115	50	220	6	8	15	3.90	4.0422	0.02841
19	115	50	210	6	8	20	3.22	3.9907	0.023273
20	115	60	210	6	8	20	3.16	2.8204	0.034727
21	115	50	230	6	8	20	3.92	4.3375	0.021628
22	110	60	220	8	6	20	2.70	1.3201	0.067252
23	115	60	210	6	4	20	3.15	2.5360	0.038284
24	115	50	220	6	4	15	3.40	3.5287	0.032924
25	115	60	220	6	4	15	3.45	2.8247	0.036098
26	110	55	210	6	6	15	3.11	2.3693	0.041732
27	115	50	230	6	4	20	3.60	3.8326	0.025915
28	120	55	230	6	6	25	4.10	4.9471	0.019705
29	115	50	220	6	4	25	3.28	4.0592	0.02502
30	115	50	220	6	8	25	3.35	4.5012	0.02117
31	115	55	220	6	6	20	3.27	3.4203	0.028938
32	115	55	210	8	6	15	3.24	2.8872	0.03453
33	120	55	210	6	6	25	3.26	4.7520	0.022605
34	120	50	220	8	6	20	3.80	4.7627	0.023984
35	115	55	220	6	6	20	3.31	3.3898	0.029125
36	110	50	220	4	6	20	3.21	2.7943	0.028558
37	120	50	220	4	6	20	3.68	4.9401	0.017
38	115	55	220	6	6	20	3.29	3.4625	0.029334
39	115	55	220	6	6	20	3.30	3.5244	0.030008
40	115	55	210	8	6	25	2.74	3.1875	0.035137
41	110	55	210	6	6	25	2.35	2.0324	0.049273
42	120	55	220	4	4	20	3.60	4.8740	0.016342
43	115	55	210	4	6	15	3.31	3.4442	0.019694
44	115	55	230	8	6	15	3.38	3.1812	0.033447
45	115	55	230	8	6	25	3.45	3.2953	0.033671
46	110	50	220	8	6	20	3.04	2.3415	0.042653
47	110	55	230	6	6	25	2.76	2.1786	0.049408
48	115	60	220	6	4	25	3.19	2.6799	0.040717
49	120	55	210	6	6	15	3.60	4.0618	0.024827
50	115	60	220	6	8	25	3.10	2.6801	0.038757
51	115	60	230	6	8	20	3.50	2.6011	0.035054
52	110	55	230	6	6	15	3.19	2.5005	0.046565
53	110	55	220	8	4	20	2.55	2.2277	0.052092
54	120	55	220	8	4	20	3.66	4.8868	0.024889

WF: wire feed; WT: wire tension; MRR: material removal rate; WC: wire consumption.

Results and discussions

Modelling and analysis

For analysis of data, checking the adequacy of model, test for significance of model and lack of fit test are required. ³⁴ For this purpose, ANOVA is performed for Ra, MRR, WC and MPCI considering both single-objective and multi-objective problems. In ANOVA tables, a p-value for the model terms that are less than 0.05 (i.e. 95% confidence level) indicates that the proposed models are considered to be statistically significant. ⁴⁰ The p-value follows the same for the factors and their interaction effect. A statistical tool ‘Design Expert (DX7)’ has been utilized to analyse the experimental data.

RSM single-objective models are involved to predict each response in any combination of parameters. The MPCI values originated from grey–fuzzy reasoning were validated using RSM model to check the adequacy of the grey–fuzzy model. Further results are checked by confirmation experiments.

Analysis of average SR through RSM model

The developed model was well satisfied as adequacy measured by R², adjusted R² and predicted R² gives satisfactory results. The parameters whose p-value is less than 0.05 are statistically significant. It can be seen in Table 3 that T_on is the most significant term contributing 57.40% followed by Ip contributing 15.03% and SV contributing 9.65% explaining the average SR (Ra). It is observed from Figure 1 and Table 3 that Ra increases with the increase in T_on and Ip values and decreases with the increase in SV value. Ra is non-linearly varied with the pulse off time. These variables have maximum impact on Ra because the energy consumed for the single pulse is the product of current, voltage and pulse on time. At high T_on and moderate T_off, an ionization and deionization of dielectric fluid are improved. The shape of the craters is uneven and also bigger in size and deep. Whereas at low T_on and T_off, the spark is uniform and craters are small and continuous which causes very fine surface. Besides this, the interactions of Ip and SV, T_on and T_off are significant for Ra. Other parameters and their interactions have minor effect on Ra as shown in Table 3. Equation (9) shows that Ra is the most non-linear complex model compared to other models. The R² value 0.9886 indicated better general ability and accuracy of the model. The adjusted R² measures the model quality and its percentage of variability. The adjusted R² 0.9832 implied that 98.32% of total variability of Ra was explained. The predicted R² 0.9701 explained that the fitted model has 97.01% variability in predicting new response values. Figure 2 displays the comparison of each observed value with the predicted value calculated from the model. It has been found that the model is fairly well fitted with the experimental values.

Table 3.

Results of ANOVA for RSM of surface roughness.

Source	Sum of squares	F-value	p-value (p > F)
Model	7.96	209.55	<0.0001*
A–Ton	4.59	2414.67	<0.0001*
B–Toff	0.1	52.72	<0.0001*
C–Ip	1.19	628.08	<0.0001*
D–WF	3.38E–4	0.18	0.6761
E–WT	0.16	82.59	<0.0001*
F–SV	0.85	446.33	<0.0001*
AB	0.14	72.58	<0.0001*
AC	0.058	30.44	<0.0001*
AD	0.083	43.53	<0.0001*
AE	0.022	11.61	0.0017*
AF	0.076	40.05	<0.0001*
BC	0.047	24.5	<0.0001*
BE	0.051	26.66	<0.0001*
CF	0.17	90.7	<0.0001*
DF	0.031	16.46	0.0003*
EF	0.063	33.19	<0.0001*
B2	0.18	95.05	<0.0001*
C2	0.014	7.14	0.0116*
D2	0.052	27.62	<0.0001*
Residual	0.063
Lack of fit	0.06	3.99	0.064
Pure error	2.68E–3
Correlation total	8.02
R2	Adjusted R2	Predicted R2	*Significant
0.9886	0.9832	0.9701

WF: wire feed; WT: wire tension.

Figure 1.

Main effect plots for surface roughness (Ra).

Figure 2.

Predicted versus actual results of SR.

After eliminating the non-significant terms using backward elimination method, the final regression equation in terms of actual form for Ra is given as follows

$$\begin{array}{c}\mathrm{Average\; roughness\; Ra}(\mathrm{\mu }\mathrm{m})\hfill \\ =90.4-1.014{\mathrm{T}}_{\mathrm{on}}-0.7942{\mathrm{T}}_{\mathrm{off}}-0.0440\mathrm{Ip}-0.7420\mathrm{WF}+0.947\mathrm{WT}-0.9777\mathrm{SV}+0.001181{\mathrm{T}}_{\mathrm{on}}*{\mathrm{T}}_{\mathrm{on}}\hfill \\ +0.005331{\mathrm{T}}_{\mathrm{off}}*{\mathrm{T}}_{\mathrm{off}}-0.000282\mathrm{Ip}*\mathrm{Ip}-0.02137\mathrm{WF}*\mathrm{WF}-0.005119\mathrm{WT}*\mathrm{WT}+0.005250{\mathrm{T}}_{\mathrm{on}}*{\mathrm{T}}_{\mathrm{off}}\hfill \\ +0.002000{\mathrm{T}}_{\mathrm{on}}*\mathrm{Ip}+0.007188{\mathrm{T}}_{\mathrm{on}}*\mathrm{WF}-0.00{525\mathrm{T}}_{\mathrm{on}}*\mathrm{WT}+0.004500{\mathrm{T}}_{\mathrm{on}}*\mathrm{SV}-0.001625{\mathrm{T}}_{\mathrm{off}}*\mathrm{Ip}\hfill \\ -0.005625{\mathrm{T}}_{\mathrm{off}}*\mathrm{WT}-0.000850{\mathrm{T}}_{\mathrm{off}}*\mathrm{SV}+0.000938\mathrm{Ip}*\mathrm{WT}+0.002225\mathrm{Ip}*\mathrm{SV}\hfill \\ +0.00813\mathrm{WF}*\mathrm{WT}+0.00625\mathrm{WF}*\mathrm{SV}-0.00937\mathrm{WT}*\mathrm{SV}\hfill \end{array}$$

(9)

Analysis of metal removal rate through RSM model

The fit summary suggests that the quadratic model is statistically significant for analysis of MRR. The reduced ANOVA for quadratic model is reported in Table 4. The model F-value 253.72 indicates that there is only 0.01% chance that a larger ‘model F-value’ could occur due to noise. F-value also helps to find significant factors and their respective ranks. Most insignificant model terms are eliminated by manual elimination of keeping alpha 0.05 (i.e. 95% confidence level). Table 4 reports that the lack of fit value is 3.72, which shows that lack of fit is not significant relative to the pure error as it is desired. The R² value for the model is calculated as 0.9922, which shows that 99.22% of the variation for metal removal rate is attributed to process variables. The predicted R² and adjusted R² values are 0.9875 and 0.9765, respectively, which are good for model. This is an indication of better general ability and accuracy of polynomial model. The control parameters T_on and T_off on MRR are found to be statistically more significant than other by contributing 74.08% and 18.20% of total contribution, respectively. From Figure 3 and Table 4, it is understood that the MRR is directly proportional to pulse on time and inversely proportional to pulse off time. The plasma channel becomes denser and wider when pulse current is applied for long time. At shorter T_off, the number of sparks striking on the surface will be more. This phenomenon may cause higher MRR. It is also observed from the figure and table that the MRR slightly varies with the variation in peak current, WF and WT within the range, while it remains almost constant for the variation in servo voltage. The comparison between the predicted value and observed value shown in Figure 4 proves the adequacy of the model.

Table 4.

Results of ANOVA for MRR with quadratic versus 2FI RSM model.

Source	Sum of squares	F-value	p-value (p > F)
Model	44.41	253.72	<0.0001*
A–Ton	33.14	3218.4	<0.0001*
B–Toff	8.14	790.29	<0.0001*
C–Ip	0.240	22.99	<0.0001*
D–WF	0.38	37.13	<0.0001*
E–WT	0.28	27.21	<0.0001*
F–SV	0.057	5.5	0.0246*
AE	0.28	26.83	<0.0001*
AF	0.41	39.43	<0.0001*
BE	0.2	19.19	<0.0001*
BF	0.22	21.38	<0.0001*
DE	0.16	15.69	0.0003*
DF	0.08	7.73	0.0086*
B2	0.6	58.75	<0.0001*
C2	0.14	13.85	0.0007*
E2	0.26	24.91	<0.0001*
Residual	0.37
Lack of fit	0.36	3.72	0.0733
Pure error	0.015
R2	Adjusted R2	Predicted R2	*Significant
0.9922	0.9875	0.9765

WF: wire feed; WT: wire tension.

Figure 3.

Main effect plots for material removal rate (MRR).

Figure 4.

Predicted versus actual results of MRR.

After eliminating the non-significant terms, the final mathematical regression equation obtained for MRR is given as follows: the MRR within this range of parameter can be predicted through this model

$$\begin{array}{c}\mathrm{Material\; removal\; rate}{(\mathrm{mm}}^3/\min )\hfill \\ =-121.8+0.1582{\mathrm{T}}_{\mathrm{on}}+1.466{\mathrm{T}}_{\mathrm{off}}+0.582\mathrm{Ip}+0.108\mathrm{WF}+2.416\mathrm{WT}-0.722\mathrm{SV}-0.00921{\mathrm{T}}_{\mathrm{off}}*{\mathrm{T}}_{\mathrm{off}}\hfill \\ -0.001091\mathrm{Ip}*\mathrm{Ip}-0.01487\mathrm{WF}*\mathrm{WF}+0.036667\mathrm{WT}*\mathrm{WT}-0.01735{\mathrm{T}}_{\mathrm{on}}*\mathrm{WT}+0.00901{\mathrm{T}}_{\mathrm{on}}*\mathrm{SV}\hfill \\ -0.001679{\mathrm{T}}_{\mathrm{off}}*\mathrm{Ip}-0.01111{\mathrm{T}}_{\mathrm{off}}*\mathrm{WT}-0.00664{\mathrm{T}}_{\mathrm{off}}*\mathrm{SV}-0.03243\mathrm{WF}*\mathrm{WT}+0.00998\mathrm{WF}*\mathrm{SV}\hfill \end{array}$$

(10)

Analysis of WC through RSM model

The fit summary suggests that the quadratic model is statistically significant for analysis of WC. The reduced ANOVA table for quadratic model of WC is reported in Table 5. For the prediction of WC, quadratic model is suggested; non-significant terms are removed by manual elimination of keeping alpha 0.05. The model is significant as p-value is less than 0.0001 and model F-value is 183.42. The ‘lack of fit F-value’ is 2.49 implying that the lack of fit is not significant relative to the pure error. Non-significant lack of fit is good. The R² value and the adjusted R² value for the model are 0.9886 and 0.9832, respectively, which shows that 98.86% of the variation for WC is attributed to process parameters, which is desirable. This is an indication of better general ability and accuracy of polynomial model. The predicted R² value 0.9701 is in reasonable agreement with adjusted R² value. It is observed from ANOVA table that the T_on has highest impact, exhibiting contribution of 57.32% followed by WF (18.52% contribution) and T_off (11.42% contribution) on WC. It is clear from Figure 5 and Table 5 that WC is inversely proportional to pulse on time and directly proportional to pulse off time and WF rate. Because when pulse on time is more, more number of sparks will be generated between work and tool electrode, causing more MRR and tool wear. At low WF, the rate of erosion in wire is more. Due to erosion, the wire strength will be reduced which leads to wire breaks. Similarly, when WF or pulse off time is more, more wire will be passed through and consumed, without any spark. So, the MRR will be less as compared to tool consumption. This phenomenon results in less utilization of consumed wire. Servo voltage and peak current have minor effect on WC. Figure 6 displays the comparison between the experimental value and calculated predicted value. It can be seen that the regression model is fairly well fitted with the experimental values.

Table 5.

Results of ANOVA for RSM of wire consumption.

Source	Sum of squares	F-value	p-value (p > F)
Model	5966.57	183.42	<0.0001*
A–Ton	3491.47	1824.65	<0.0001*
B–Toff	681.31	356.06	<0.0001*
C–Ip	3.09	1.61	0.2119
D–WF	1091.41	570.37	<0.0001*
E–WT	57.82	30.22	<0.0001*
F–SV	6.36	3.32	0.0765
AB	164.33	85.88	<0.0001*
AC	17.49	9.14	0.0046*
AD	57.66	30.13	<0.0001*
AF	23.69	12.38	0.0012*
BD	47.91	25.04	<0.0001*
BF	83.5	43.64	<0.0001*
A2	193.92	101.34	<0.0001*
B2	48.23	25.21	<0.0001*
D2	48.15	25.16	<0.0001*
F2	21.24	11.1	0.002*
Residual	68.89
Lack of fit	64.69	2.49	0.1566
Pure error	4.2
R2	Adjusted R2	Predicted R2	*Significant
0.9886	0.9832	0.9701

WF: wire feed; WT: wire tension.

Figure 5.

Main effect plots for wire consumption (WC).

Figure 6.

Predicted versus actual results of wire consumption.

After eliminating the non-significant terms, the final mathematical regression equation obtained for WC is given as follows

$$\begin{array}{c}\mathrm{Wire\; consumption}(\mathrm{kg})\hfill \\ =0.741-0.02273{\mathrm{T}}_{\mathrm{on}}+0.00938{\mathrm{T}}_{\mathrm{off}}+0.00364\mathrm{Ip}+0.01833\mathrm{WF}-0.000818\mathrm{WT}-0.00132\mathrm{SV}\hfill \\ +0.000173{\mathrm{T}}_{\mathrm{on}}*{\mathrm{T}}_{\mathrm{on}}+0.000077{\mathrm{T}}_{\mathrm{off}}*{\mathrm{T}}_{\mathrm{off}}-0.000518\mathrm{WF}*\mathrm{WF}+0.000055\mathrm{SV}*\mathrm{SV}\hfill \\ -0.000181{\mathrm{T}}_{\mathrm{on}}*{\mathrm{T}}_{\mathrm{off}}-0.000032{\mathrm{T}}_{\mathrm{on}}*\mathrm{Ip}-0.000177{\mathrm{T}}_{\mathrm{on}}*\mathrm{WF}-0.000064{\mathrm{T}}_{\mathrm{on}}*\mathrm{SV}\hfill \\ +0.000245{\mathrm{T}}_{\mathrm{off}}*\mathrm{WF}+0.000129{\mathrm{T}}_{\mathrm{off}}*\mathrm{SV}-0.000089\mathrm{WF}*\mathrm{SV}\hfill \end{array}$$

(11)

Multi-objective optimization

The MRR has been considered as ‘higher the better’, and Ra and WC have been considered as ‘lower the better’. The normalized S/N ratios are reported in Table 6. The larger value of normalized result can indicate the better performance characteristics; best results will be equal to 1.

Table 6.

Normalized S/N ratios.

S. no.	Normalized S/N ratio
	Ra	MRR	WC
1	0.5733	0.6012	0.4401
2	0.3945	0.7416	0.5412
3	0.067	0.8173	0.5013
4	0.3101	0.5588	0.372
5	0.1318	0.9264	0.6296
6	0.0221	0.9272	0.7323
7	0.1942	0.5889	0.4335
8	0.5553	0.7383	0.5634
9	0.3945	0.7394	0.4961
10	0.2843	0.7711	0.7476
11	0.4009	0.727	0.6444
12	0.6534	0.4504	0.2143
13	0.1271	0.845	0.9563
14	0.2537	0.9873	1
15	0.6916	0.1783	0.2813
16	0.8184	0.6861	0.618
17	0.7176	0.3538	0.3781
18	0.0899	0.8471	0.5175
19	0.4341	0.8374	0.6373
20	0.4679	0.5747	0.3969
21	0.0807	0.9005	0.6813
22	0.7241	0	0
23	0.4736	0.4942	0.3383
24	0.3364	0.7442	0.4289
25	0.3101	0.5758	0.3737
26	0.4966	0.4427	0.2866
27	0.2337	0.8068	0.5727
28	0	1	0.7372
29	0.4009	0.8503	0.5938
30	0.363	0.9285	0.6941
31	0.3945	0.7206	0.5064
32	0.423	0.5924	0.4003
33	0.4119	0.9695	0.6547
34	0.1365	0.9712	0.6192
35	0.3945	0.7139	0.5026
36	0.4397	0.5676	0.5144
37	0.1942	0.9989	0.8259
38	0.3945	0.7299	0.4983
39	0.3945	0.7433	0.4846
40	0.7241	0.6673	0.3899
41	1	0.3266	0.1868
42	0.2337	0.9887	0.8496
43	0.3846	0.7259	0.7375
44	0.347	0.6658	0.4195
45	0.3101	0.6924	0.4155
46	0.5375	0.4338	0.2735
47	0.7111	0.3792	0.1852
48	0.4509	0.536	0.3013
49	0.2337	0.8507	0.5984
50	0.5023	0.536	0.331
51	0.2843	0.5134	0.3913
52	0.4509	0.4835	0.2208
53	0.8533	0.3961	0.1534
54	0.204	0.9907	0.5969

S/N: signal-to-noise; MRR: material removal rate; WC: wire consumption.

Calculation of grey relational coefficient

The grey relational coefficients for those performance characteristics are evaluated using equation (4) and reported in Table 7. The deviation sequence, ${\Delta }_{\mathrm{oi}}(m)$ , is determined using equation (5).

Table 7.

Computed grey coefficient (ξ) and MPCI.

ξ for Ra	ξ for MRR	ξ for WC	MPCI
0.54	0.556	0.472	0.549
0.452	0.65	0.502	0.48
0.349	0.732	0.501	0.524
0.42	0.531	0.443	0.419
0.365	0.872	0.574	0.545
0.338	0.873	0.651	0.546
0.383	0.549	0.469	0.389
0.529	0.656	0.534	0.541
0.451	0.649	0.499	0.475
0.411	0.686	0.665	0.579
0.455	0.647	0.584	0.549
0.591	0.476	0.389	0.552
0.364	0.763	0.92	0.58
0.401	0.975	1	0.727
0.619	0.378	0.41	0.497
0.734	0.614	0.567	0.677
0.639	0.436	0.446	0.551
0.355	0.766	0.509	0.532
0.469	0.755	0.58	0.584
0.484	0.54	0.453	0.477
0.352	0.834	0.611	0.525
0.644	0.333	0.333	0.471
0.487	0.497	0.43	0.479
0.43	0.662	0.467	0.455
0.42	0.541	0.444	0.419
0.498	0.473	0.412	0.459
0.395	0.721	0.539	0.561
0.333	1	0.655	0.651
0.455	0.77	0.552	0.583
0.44	0.875	0.62	0.583
0.452	0.647	0.499	0.473
0.464	0.551	0.455	0.455
0.46	0.943	0.592	0.605
0.367	0.946	0.568	0.571
0.45	0.648	0.498	0.474
0.472	0.536	0.507	0.475
0.383	0.998	0.742	0.663
0.452	0.655	0.496	0.483
0.45	0.661	0.492	0.488
0.644	0.6	0.45	0.597
1	0.426	0.381	0.667
0.395	0.978	0.769	0.635
0.448	0.646	0.656	0.559
0.434	0.599	0.463	0.43
0.42	0.619	0.461	0.439
0.519	0.469	0.408	0.485
0.634	0.446	0.38	0.574
0.477	0.519	0.417	0.463
0.395	0.77	0.555	0.566
0.501	0.519	0.428	0.502
0.411	0.507	0.451	0.412
0.477	0.492	0.391	0.468
0.773	0.453	0.371	0.583
0.386	0.982	0.554	0.559

MRR: material removal rate; WC: wire consumption; MPCI: multiple performance characteristics index.

Calculation of MPCI

MATLAB tool was used for obtaining the grey–fuzzy output (MPCI) from the responses such as Ra, MRR and WC (Figure 7). By Mamdani inference, the fuzzy linguistic values and their membership values for the outputs are obtained. The quality characteristic evaluation plan that has been designed as triangular membership function is applied for the two grey coefficients, each with seven membership functions and a typical plot as shown in Figure 8. The MPCI is also divided into seven number of membership functions. Seven fuzzy sets, very low (VL), low (L), fairly low (FL), medium (M), fairly high (FH), high (H) and very high (VH), are postulated for the Ra, MRR, WC and MPCI within the universe of discourse of [0, 1]. For activating the fuzzy inference system, a total of 370 possible numbers of fuzzy rules are created according to expert’s experiences (Figure 9). Graphical representation of the fuzzy logic reasoning procedure for prediction of MPCI using the MATLAB software is shown in Figure 10. The fuzzy inference system is evaluated to predict the MPCI for all 54 experiments which has been reported in Table 7. The highest value of MPCI 0.727 (expt no. 14) indicates the optimum combination of parameter setting within the experiments. But the parameter setting is not the optimum parameter combination within the full range of study in order to produce maximum MRR and minimum Ra and WC. Since the highest MPCI value is acceptable, the S/N ratio plots are introduced in Figure 11 to find optimal parametric combination within the range and distinguish the variations for each of the process parameters on MPCI. The optimal setting becomes T_on3 − T_off2 − Ip1 − WF1 − WT2 − SV3.

Figure 7.

Proposed fuzzy inference system.

Figure 8.

Membership functions for normalized S/N ratio of SR, MRR and WC.

Figure 9.

Fuzzy rule base (MATLAB).

Figure 10.

Fuzzy reasoning rule base output (MATLAB).

Figure 11.

Evaluation of optimal setting (S/N ratio plot of MPCIs).

Analysis of MPCI

Equation (12) shows the modelling of MPCI using RSM. The model F-value is 13.89 (p < 0.0001) implying that the grey–fuzzy model of MPCI is significant and also validates the model accuracy

$$\begin{array}{cc}\multicolumn{1}{c}{\mathrm{MPCI}}& =24.09-0.477{\mathrm{T}}_{\mathrm{on}}+0.1142{\mathrm{T}}_{\mathrm{off}}-0.002138\mathrm{Ip}+0.288\mathrm{WF}+0.00269\mathrm{WT}\hfill \\ \multicolumn{1}{c}{}& +0.00861\mathrm{SV}+0.002174{\mathrm{T}}_{\mathrm{on}}*{\mathrm{T}}_{\mathrm{on}}-0.001109{\mathrm{T}}_{\mathrm{off}}*{\mathrm{T}}_{\mathrm{off}}-0.00266{\mathrm{T}}_{\mathrm{on}}*\mathrm{WF}\hfill \end{array}$$

(12)

The MPCIs obtained from equations (4) and (12) are analysed using ANOVA. ANOVA is used to reveal the influence of factors on MPCI. As per ANOVA reported in Table 8 and supported S/N ratio plot in Figure 11, four most influencing factors on overall performance are SV, T_off, T_on and WF. The contributions of SV, T_off, T_on, WF and Ip on MPCI are 15.24%, 12.24%, 10.10%, 9.84% and 3.76%, respectively. Strong interaction is found in second-order polynomial of T_on (contribution of 16.39%). Good interaction is observed between pulse on time–wire feed rate and pulse current–servo voltage for the reduction in MPCI.

Table 8.

Results of ANOVA for MPCI.

Source	Sum of squares	F-value	p-value (p > F)
Model	0.243	13.89	<0.0001*
A–Ton	0.029	17.81	0.0001*
B–Toff	0.036	21.59	<0.0001*
C–Ip	0.011	6.63	0.014*
D–WF	0.029	17.34	0.0001*
E–WT	0.0007	0.42	0.521
F–SV	0.0445	26.86	0.0001*
AD	0.0113	6.82	<0.012*
A2	0.0315	22.84	0.0001*
B2	0.0086	5.94	<0.019*

WF: wire feed; WT: wire tension. *Significant values.

Confirmation experiment

In order to verify the accuracy of the RSM model, the confirmation experiments are reported in Table 9. The optimum combination of process parameters according to grey–fuzzy method is also checked by confirmation experiment and the results are reported in Table 10. The error percentages are within permissible limits which confirmed the adequacy of each model. Figure 12 shows the 3D micron-scale surface topography and the roughness profile of the optimum machining condition achieved from the confirmation experiment.

Table 9.

Confirmation test for single-objective optimization problem using RSM model.

Process parameter	Experimental			Predicted			Error (%)
Ton-Toff - Ip-WF-WT-SV	Ra	MRR	WC	Ra	MRR	WC	Ra	MRR	WC
120-60-230-8-8-25	3.93	3.86	27.12	4.10	3.6	27.69	4.32	−7.22	2.06
115-55-220-6-6-20	3.27	3.52	30.05	3.30	3.5	28.52	0.09	−0.57	−5.36
110-50-210-4-4-15	2.91	2.3	34.24	3.10	2.16	30.46	6.52	−6.48	−12.4
118-58-215-5-7-22	3.94	3.68	20.73	3.34	3.81	21.88	−3.3	3.53	5.54

WF: wire feed; WT: wire tension; MRR: material removal rate; WC: wire consumption.

Table 10.

Optimal solutions as obtained by grey–fuzzy algorithm for multi-objective optimization problem.

Optimum condition (MPCI 0.7464)						Experimental optimum performance			Predicted optimum performance
Ton	Toff	Ip	WF	WT	SV	Ra	MRR	WC	Ra	MRR	WC
120	55	210	4	6	25	3.16	4.775	17.325	3.01	4.701	16.749

MPCI: multiple performance characteristics index; WF: wire feed; WT: wire tension; MRR: material removal rate; WC: wire consumption.

Figure 12.

3D micron-scale surface topography (optimum).

Conclusion

Based on the experimental study and data analysis, the effects of process parameters (such as T_on, T_off, current, WT, WF and servo voltage) on each performance characteristics (Ra, MRR and WC) and overall performance (MPCI) are systematically investigated using RSM, grey–fuzzy system and ANOVA. RSM successfully predicts the average roughness, MRR and WC. Grey–fuzzy analysis converts multi-response into single MPCI. Fuzzy logic and RSM are successfully applied for optimization of process parameter and non-linear modelling of responses. It was found that T_on of 120 mu (1.1 μs), T_off of 55 mu (36 μs), pulse current of 210 A, WF of 4 m/min, WT of 6 mu (0.7 kg) and voltage of 25 V are the optimal combination of WEDM parameters based on high value of MPCI (0.7464). The optimum parameters can be directly used by the end user. At this level of parameter, sparking is distributed uniformly throughout the surface; as a result, uniform surface is produced with high MRR and low WC. ANOVA confirmed the adequacy of each model. It revealed that the SV, T_off, T_on and WF are the most influential parameters on MPCI. The confirmation tests showed that the mathematical model can successfully predict the performance parameter as the errors between the experimental and predicted values of Ra, MRR and WC are within considerable limit.