# Integration of Fuzzy AHP and Fuzzy TOPSIS Methods for Wire Electric Discharge Machining of Titanium (Ti6Al4V) Alloy Using RSM

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## Abstract

**:**

_{on}), pulse-off time (T

_{off}), and current—were chosen, each having three levels to monitor the process response in terms of cutting speed (V

_{C}), material removal rate (MRR), and surface roughness (SR). To assess the relevance and significance of the models, an analysis of variance was carried out. The optimal process parameters after integrating fuzzy AHP coupled with fuzzy TOPSIS approach found were T

_{on}= 40 µs, T

_{off}= 15 µs, and current = 2A.

## 1. Introduction

_{on}, T

_{off}, spark gap voltage, peak current, T

_{on}× T

_{off}, T

_{on}× IP, T

_{on}× wire tension, and spark gap voltage × wire tension have significant effects on cutting speed. Furthermore, a mathematical model of dimensional deviation suggests that the main effects of T

_{on}, T

_{off}, SV, IP, and WT, and interaction effects between T

_{on}and SV, T

_{on}and IP are statistically significant for the analysis. Kavimani et al. [25] proposed a mathematical model for predicting the response data of MRR and surface roughness by using regression analysis with orthogonal array. The mathematical model proposed by Bose and Nandi [26] is based on two factors and four levels design of experiments on output response, such as surface roughness. They used desirable gray relational analysis algorithm for optimization of the WEDM process and model building. Daniel et al. [27] analyzed the effects of several WEDM process parameters on MRR and developed a mathematical model using regression analysis for predicting the response.

_{on}), wire tension, servo voltage, and pulse-off time (T

_{off}) were studied on the kerf width, surface roughness (SR), and MRR of the material. An analysis of variance (ANOVA) was conducted to check the impact of the process variables on the desired responses. Chaudhari et al. [29] attempted machining of Ni55.8Ti super-elastic shape memory alloy with the WEDM process. To exhibit the viability for industrial applications, a scientific approach consisting of RSM and HTS algorithm was planned and prepared for optimization. Saedon et al. [30] studied optimization of kerf and MRR during cutting of titanium (Ti6Al4V) alloy using WEDM. The experiments were performed with varying peak current, T

_{off}, wire tension, and wire feed. The design of the process parameters for the experiment was determined using Taguchi’s L9 design. The GRA approach was utilized for emphasizing the study for multiple performance characteristics. Payal et al. [31] employed the Taguchi fuzzy integration for parametric optimization of EDM process with multiple response measures. A fuzzy model was formed through which the optimal blend of parameters was obtained on the basis of multi-performance fuzzy index values.

_{on}, wire feed rate (WF), T

_{off}, and wire type as input parameters. A similar study was performed using the AHP–TOPSIS hybrid approach during WEDM by Nayak and Mahapatra [39]. The current research on optimization is focused on developing a hybrid approach of optimization by integrating different MCDM techniques with a fuzzy approach. The fuzzy approach is a method of solving problems that are associated with uncertainty and vagueness. Different types of uncertainty can be observed in different optimization and decision-making problems. Chou et al. [40] used the fuzzy AHP and fuzzy TOPSIS to assess the performance of human resources in science and technology (HRST) in Southeast Asian countries. A fuzzy AHP approach was implemented to decide the preference weights for various performance measures while fuzzy TOPSIS was used to identify the best tread-off alternatives to accomplish the ideal HRST levels. They concluded that countries such as Singapore, South Korea, and Taiwan have better HRST performances compared to other Southeast Asian countries. Sirisawat and Kiatcharoenpol [41] employed the fuzzy AHP and fuzzy TOPSIS techniques to classify the reverse logistics barriers and also to prioritize and rank the solutions to implementing reverse logistics in the electronics industry. A hybrid decision-making method with a fuzzy approach was also found effective in deciding the best alternative in the manufacturing field. Roy and Dutta [42] studied the working of the EDM process considering duty cycle, current, T

_{on}, and gap voltage as controllable parameters. They used Taguchi’s L9 technique for experimental design and used integrated fuzzy AHP and fuzzy TOPSIS methods to determine the optimal set of controllable parameters. Furthermore, ANOVA analysis of closeness coefficient index (CC

_{i}) revealed that the current was the highest contributing factor among the selected parameters.

_{on}, T

_{off}, and current during the WEDM of Ti6Al4V alloy. Cutting speed (V

_{C}), MRR, and SR were considered for the analysis as output response variables. The experiments were systematically designed using central composite design (CCD) of RSM and mathematical models were developed between responses and input parameters. The appropriateness of developed models was checked using ANOVA by analyzing the R-Seq values, lack-of-fit, and residual plots. Furthermore, the weights were assigned to considered criterion using fuzzy AHP, and optimal process parameter conditions were predicted using fuzzy TOPSIS. The confirmatory study was performed to verify the results of optimization.

## 2. Materials and Methods

#### 2.1. Experimental Methodology

^{3}) and molybdenum wire (180 µm diameter) as the tool electrode. The dielectric fluid utilized was deionized water. Table 1 shows the chemical composition of the selected work material of Ti6Al4V. T

_{on}, T

_{off}, and current were selected as input process parameters based on recent literature to investigate their effects on WEDM machining in terms of cutting speed, MRR, and SR. Full factorial CCD of RSM was selected to design the experimental plan. CCD is very popular amongst multiple variants of RSM approaches because it offers great flexibility and allows sequential operations and effectiveness by providing the optimum solution in a minimum number of iterations. CCD design of RSM for three factors at three levels was implemented to prepare the experimental matrix. The levels of factors were selected based on preliminary trials and literature study. Table 2 shows the input parameters of the WEDM process at the three levels. Table 3 shows the full factorial CCD composed of the six axial and central points, and eight factorial points. During each experimental run, the wire was fed along a width direction of the work material. Each experiment was repeated three times; the average values of the three trials were reported and considered for the analysis. Minitab 17 software was utilized for the RSM design and analyzing the experimental data.

_{initial}− wt.

_{final})/(ρ ∗ t)

_{initial}is the weight of the workpiece measured before the cut and wt.

_{final}is the weight of workpiece measured after the cut. ρ is the density of the titanium alloy, 4.42 gm/cm

^{3}. t is the cutting time measured for a particular cut in one minute.

#### 2.2. Fuzzy Analytical Hierarchy Process

- Step 1:
- Construct the various leveled structure of objective, criterion, and alternatives of the problem.
- Step 2:
- Construct a pairwise comparison matrix from the criteria/options available. Furthermore, assign linguistic terms using Figure 1 to the pairwise comparisons collected from decision makers. Convert linguistic terms into fuzzy numbers using Table 4. The generalized pairwise comparison matrix will be of the form shown in Equation (3).$$\tilde{A}=\left(\begin{array}{ccc}1& \dots & {\tilde{y}}_{1n}\\ \vdots & \ddots & \vdots \\ \frac{1}{{\tilde{y}}_{1n}}& \cdots & 1\end{array}\right)$$
- Step 3:
- Fuzzification is used to convert the linguistic term into a membership term. The fuzzification of the linguistic term can be possible using various functions such as triangular, bell-shaped, and trapezoidal functions. For this study, we used the triangular membership function, as shown in Figure 2. The assumed fuzzy numbers are shown in Equations (4) and (5).$$X=\left(r,s,t\right)$$$$Y=\left(l,m,n\right)$$Fuzzy weights can be found using fuzzy addition and multiplication [45]. The generalized fuzzy addition and fuzzy multiplication formulas are expressed by Equations (6) and (7).Fuzzy addition:$$\tilde{X}\oplus \tilde{Y}=(\mathrm{r},\mathrm{s},\mathrm{t})\oplus (\mathrm{l},\mathrm{m},\mathrm{n})=(\mathrm{r}+l,\mathrm{s}+m,\mathrm{t}+n)$$Fuzzy multiplication:$$\tilde{X}\otimes \tilde{Y}=(\mathrm{r},\mathrm{s},\mathrm{t})\otimes (\mathrm{l},\mathrm{m},\mathrm{n})=(\mathrm{r}\times l,\mathrm{s}\times m,\mathrm{t}\times n)$$
- Step 4:
- Determine the fuzzy mean geometric value (FMGV) of each criteria using the geometric mean method. Equation (8) can be used for calculating FMGV. The fuzzy weights can be determined by using Equation (9).$${\tilde{\mathrm{r}}}_{j}={({\tilde{y}}_{j1}\otimes {\tilde{y}}_{j2}\otimes \dots \otimes {\tilde{y}}_{jn})}^{\frac{1}{n}}$$$${\tilde{w}}_{j}={\tilde{r}}_{j}\otimes {({\tilde{\mathrm{r}}}_{1}\oplus {\tilde{\mathrm{r}}}_{2}\oplus \dots \oplus {\tilde{\mathrm{r}}}_{n})}^{-1}$$

#### 2.3. Fuzzy TOPSIS

- Step 1:
- Normalization of response: The normalization is important for converting measured outputs into the fuzzy number. The process of normalization was carried out considering the output based on the benefit criteria or the cost criteria. The V
_{C}and MRR were normalized using the benefit criteria using Equation (10), whereas SR was normalized using the cost criteria using Equation (11).For benefit criteria:$${r}_{ij}\left(x\right)=\frac{{x}_{ij}-{x}_{min}}{{x}_{max}-{x}_{min}}$$For cost criteria:$${r}_{ij}\left(x\right)=\frac{{x}_{min}-{x}_{ij}}{{x}_{min}-{x}_{max}}$$ - Step 2:
- Fuzzification of normalized decision matrix: The decision matrix normalized in Step 1 can be converted to a fuzzified normalized decision matrix by assigning a sub-criteria grade to each alternative using Table 5 of the K membership function scale. Additionally, assign the weights to each sub-criteria grade.The weight of criteria:$${\tilde{\mathrm{w}}}_{j}^{k}=({\tilde{\mathrm{w}}}_{j1}^{k},{\tilde{\mathrm{w}}}_{j2}^{k},{\tilde{\mathrm{w}}}_{j3}^{k})$$
- Step 3:
- Calculate the weighted normalized fuzzy decision matrix: The weights obtained from fuzzy AHP are required to construct this matrix. The weighted normalized values can be calculated as:$$\tilde{V}=({\tilde{v}}_{ij})\mathrm{Where}{\tilde{v}}_{ij}={\tilde{r}}_{ij}\times {w}_{j}$$
- Step 4:
- Identify the positive ideal (V
^{+}) and negative ideal (V^{−}) solutions: The fuzzy positive ideal solutions (FPIS, V^{+}) and the fuzzy negative ideal solutions (FNIS, V^{−}) must be calculated using Equations (14) and (15).${V}^{+}=({\tilde{\mathrm{v}}}_{1}^{+},{\tilde{\mathrm{v}}}_{2}^{+},{\tilde{\mathrm{v}}}_{3}^{+})$, where:$${\tilde{v}}_{j}^{+}=\underset{i}{\mathrm{max}}\left\{{\mathrm{v}}_{ij3}\right\}$$${V}^{-}=({\tilde{\mathrm{v}}}_{1}^{-},{\tilde{\mathrm{v}}}_{2}^{-},{\tilde{\mathrm{v}}}_{3}^{-})$, where:$${\tilde{v}}_{j}^{-}=\underset{i}{\mathrm{min}}\left\{{\mathrm{v}}_{ij1}\right\}$$Consideration of the maximum and minimum of V_{ij}does not necessarily result in triangular fuzzy numbers, but we can obtain the ideal solutions as the fuzzy numbers using Equation (16).$$d\left(\tilde{p},\tilde{q}\right)=\sqrt{\frac{1}{3}\left\{{\left({p}_{1}-{q}_{1}\right)}^{2}+{\left({p}_{2}-{q}_{2}\right)}^{2}+{\left({p}_{3}-{q}_{3}\right)}^{2}\right\}}$$ - Step 5:
- Calculate separation measures: The separation measure ${d}_{i}^{+}$ is the summation of the distance of each response to the FPIS and ${d}_{i}^{-}$ is the summation of the distance of each response to the FNIS. The distance can be calculated by using the following equations.$${d}_{i}^{+}={\displaystyle \sum _{j=1}^{n}d({\tilde{\mathrm{v}}}_{ij,}{\tilde{\mathrm{v}}}_{j}^{+})};i=1,2,\dots ,m$$$${d}_{i}^{-}={\displaystyle \sum _{j=1}^{n}d({\tilde{\mathrm{v}}}_{ij,}{\tilde{\mathrm{v}}}_{j}^{-})};i=1,2,\dots ,m$$
- Step 6:
- Calculate the similarities to the ideal solution: To solve the similarities, compute the closeness coefficient CC
_{i}for each alternative [48].$$C{C}_{i}=\frac{{d}_{i}^{-}}{{d}_{i}^{+}+{d}_{i}^{-}};C{C}_{i}\in [0,1]\forall i=1,2,\dots ,n$$

## 3. Results and Discussions

#### 3.1. Regression Equations

_{C}, MRR, and SR in terms of input process variables. Equations (20)–(22) show the regression equations for V

_{C}, MRR, and SR, respectively.

**V**= 1.429 + 0.00567(T

_{C}_{on}) − 0.166(T

_{off}) + 1.613(Current) − 0.00005(T

_{on}× T

_{on}) + 0.00233(T

_{off}× T

_{off}) − 0.1699(Current × Current) + 0.000265(T

_{on}× T

_{off}) + 0.001450(T

_{on}× Current) − 0.01280(T

_{off}× Current)

**MRR**= 1.37 + 0.0153(T

_{on}) − 0.118(T

_{off}) + 1.552(Current) − 0.000111(T

_{on}× T

_{on}) + 0.00062(T

_{off}× T

_{off}) − 0.1345(Current × Current) + 0.000267(T

_{on}× T

_{off}) + 0.00183(T

_{on}× Current) − 0.01450(T

_{off}× Current)

**SR**= −1.48 + 0.1273(T

_{on}) − 0.341(T

_{off}) + 2.66(Current) − 0.000664(T

_{on}× T

_{on}) + 0.0133(T

_{off}× T

_{off}) − 0.318(Current × Current) − 0.00167(T

_{on}× T

_{off}) + 0.01204(T

_{on}× Current) − 0.0443(T

_{off}× Current)

#### 3.2. Analysis of Cutting Speed

_{C}. T

_{off}shows the highest significance for obtaining higher value of V

_{C}followed by current and T

_{on}. The p-value of the model is also less than 0.05, highlighting that the model is significant and best fitted for the selected range of process parameters. Insignificant lack-of-fit reveals the adequacy and fitness of the model [10]. The value of R

^{2}indicates that 99.02% of the variation of cutting speed is contributed by the control factors and only 0.98% of total variation cannot be described by the quadratic model. The adjacent R-squared’ and predicted R-squared values of 98.14% and 94.04%, respectively, are in reasonable agreement. Figure 3 shows the normal probability plot of residuals for V

_{C}. ANOVA results are considered to be valid depending on the analysis of these plots. We observed that the developed regression model fit well with the observed values.

_{C}. The increase in the value of T

_{on}signifies a rise in the duration of a spark, which causes the discharge energy to increase [2]. An increase in T

_{on}and current significantly increases the spark intensity, which in turn escalates the melting and vaporization of the material from workpiece [7]. Hence, an increase in both T

_{on}and current increases V

_{C}. However, a continuous decrease in the value of V

_{C}has been observed with an increase in the value of T

_{off}due to the absence of the spark during the machining [10].

#### 3.3. Analysis of MRR

_{on}, T

_{off}and current) show p-values of less than 0.05, suggesting that all are having a significant effect on MRR. The regression equation and ANOVA results of MRR revealed that MRR is significantly dependent on T

_{off}. T

_{off}is most contributing factor (41.73%) on MRR followed by T

_{on}and current. Saedon et al. [30] also reported T

_{off}as the most significant factor (58%) while investigating WEDM machining on Ti-6Al-4V alloy. The R

^{2}value of 98.29% indicates that 98.29% of the variation of MRR can be explained by the empirical model and only 1.71% of total variation cannot be described by the developed model. When predicted R

^{2}and adjusted R

^{2}values are in reasonable agreement, this confirms a strong correlation between observed and predicted values. Here, adjusted R

^{2}is 96.76%, and the predicted value of R

^{2}is 88.69%. The closeness of the values depicts a strong correlation between them. Figure 5 shows the residual plots for MRR in terms of normal probability plot versus residual, residual versus fitted values, histogram, and residual versus observation order. All the residual plots indicate that the regression model fits well with the observed values.

_{on}and current, the influence of clearance form on MRR exhibits an increasing tendency. An increase in T

_{on}and current significantly increases the spark intensity which in turn escalates the melting and vaporization of the material from the workpiece [10]. This further increases the MRR by a large amount. Saedon et al. [30] attributed increasing MRR with increasing T

_{on}and current to the reduced dynamic shear strength of Ti alloy due to higher thermal influence in the machining region. The outcome of T

_{off}on MRR shows a decreasing trend with a rise in T

_{off}because of reduced spark ejection time and less MRR [9,54]. Thus, with an increase in T

_{off}, MRR is decreasing. Moreover, the slope indicates that it has a great effect on MRR; a slight increase in T

_{off}leads to a decrease in MRR [9]. The positive dependency/correlation/relationship of MRR with T

_{on}and negative dependency with T

_{off}is also reported by Arikatla et al. [28].

#### 3.4. Analysis of SR

_{on}and current were found to be the significant input process parameters for SR, with a higher contribution of T

_{on}followed by current. The ANOVA results of SR revealed T

_{on}as the main contributing factor (50.09%). Similarly, the largest contributing effect of T

_{on}on SR was also reported by Arikatla et al. [28] A lack of fit of 3.73 with a corresponding p-value of 0.087 implies that it is not significant. The value of R

^{2}of 0.9131 indicates that 91.31% of the variation of surface roughness can be explained by the empirical model and only 8.69% of total variation cannot be described by the developed model. When predicted R

^{2}and adjusted R

^{2}values are in reasonable agreement, it indicates strong relationship in observed and predicted values. Here, adjusted R

^{2}is 0.9842 and the predicted value of R

^{2}is 0.9473. Table 10 shows the model summary for all the responses. The closeness of the values depicts a strong correlation between them. Figure 7 shows the normal probability plot of residuals for SR. The plots highlight that that developed regression model fits well with the observed values.

_{on}is directly proportional to SR and has a constantly increasing trend because the higher the T

_{on}, the higher the energy of discharge and spark intensity, resulting in poor surface texture and vice versa [55]. This indicates that with the increase in T

_{on}, the SR value also increases. The effect of T

_{off}on SR shows a reducing trend with growth in T

_{off}(inversely proportional) [50,55]. Thus, with the increase in T

_{off}, SR is decreasing as discharge energy falls with the rise in T

_{off}, leading to the plunging of the crater dimensions and hence reducing SR. The effect of current on SR shows a growing tendency with a rise in current value and vice versa. Higher SR with the high current can be explained by the fact that crater sizes are also dependent on the ionization of the dielectric fluid and ionization takes place at faster rate with a higher current [50]. Thus, crater size increases with an increase in current value, resulting in higher SR. The positive dependency of SR on T

_{on}and negative dependency on T

_{off}was also reported by Arikatla et al. [28].

_{C}and MRR are of the “higher the better” category and SR falls under the “lower the better category”. Thus, the optimum parameter settings obtained considering a single objective are shown in Table 11. It can be observed that the optimum settings are conflicting in nature when all three responses are considered together. The industry demands process parameter setting, which can result in higher productivity (higher V

_{C}and MRR) with good quality (low SR). This requires an improved means of optimization which can take care of such conflicting situations.

#### 3.5. Optimization Using Integrated Fuzzy AHP and Fuzzy TOPSIS

#### 3.5.1. Fuzzy AHP

#### 3.5.2. Fuzzy TOPSIS

_{C}and MRR are benefit criteria and SR is cost criteria. Thus, we assumed FPIS ${\mathrm{V}}_{\mathrm{j}}^{+}$ as (1, 1, 1) for V

_{C}, MRR and (0, 0, 0) for SR. Additionally, we assigned FNIS ${\mathrm{V}}_{\mathrm{j}}^{-}$ as (0, 0, 0) for V

_{C}, MRR and (1, 1, 1) for SR.

_{i}) was found for each alternative using Equation (19). The value of CC

_{i}indicates whether the alternative is nearest to theoretical FPIS and furthest from the theoretical FNIS or vice versa [57]. The highest rank is given to the alternative with the highest value of the closeness coefficient, as shown in Table 17.

_{i}values. High CC

_{i}values are always preferred. Hence, the (S/N) ratio was calculated using the “higher the better” strategy in Minitab. The obtained main effect plot of the (S/N) ratio is shown in Figure 9. The graph indicates the effect of factors’ concerning level. The optimum levels of process parameters were picked based on a higher value of the S/N ratio for the levels of factors. The optimized process parameters based on the highest values S/N ratio were a T

_{on}of 40 µs, T

_{off}of 15 µs, and current of 2A. The determined optimized process parameters correspond to alternative 12.

## 4. Conclusions

- Response surface methodology is effective for systematically designing the experiments. The mathematical relations developed between dependent and independent parameters are significant for predicting the responses at a 95% confidence interval.
- ANOVA analysis confirmed that the input parameters T
_{on}, T_{off}, and current significantly affect cutting speed, material removal rate, and surface roughness. - Fuzzy AHP can be incorporated to prioritize the responses using data collected from experts. The use of the fuzzy approach eliminates the aleatory uncertainty present in the natural language. The weights calculated using fuzzy AHP can be incorporated in fuzzy TOPSIS without bias.
- For the considered range of process parameters, the optimal process parameters for WEDM are T
_{on}= 40 µs, T_{off}= 15 µs, and current = 2A. - The confirmatory experiments proved that fuzzy logic is an effective and efficient solution for the optimization of WEDM process parameters. The proposed integrated approach of RSM, fuzzy AHP, and fuzzy TOPSIS can be further extended for different machining processes.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

AHP | Analytical hierarchy process |

ANOVA | Analysis of variance |

CC_{i} | Closeness coefficient index |

CCD | Central composite design |

CR | Consistency ratio |

FMGV | Fuzzy mean geometric value |

FPIS | Fuzzy positive ideal solutions |

FNIS | Fuzzy negative ideal solutions |

GRA | Gray relational analysis |

HTS | Heat transfer search |

MCDM | Multi-criteria decision making |

MRR | Material removal rate |

RSM | Response surface methodology |

S/N | Single to noise |

SR | Surface roughness |

TOPSIS | Technique for Order Preference by Similarity to Ideal Solution |

T_{on} | Pulse-on time |

T_{off} | Pulse-off time |

V_{C} | Cutting speed |

WF | Wire feed rate |

WEDM | Wire electrical discharge machining process |

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**Figure 1.**Membership function fuzzy scale for AHP along with linguistic variables [44].

**Figure 2.**Triangular fuzzy number [45].

C | Fe | Al | N_{2} | Cu | V | Ti |
---|---|---|---|---|---|---|

0.05 | 0.20 | 6.20 | 0.04 | 0.001 | 4.0 | Balanced |

Parameter | Symbol | Unit | Level 1 | Level 2 | Level 3 |
---|---|---|---|---|---|

Pulse-on time (T_{on}) | A | µs | 40 | 70 | 100 |

Pulse-off time (T_{off}) | B | µs | 15 | 20 | 25 |

Current | C | A | 2 | 3 | 4 |

Std. Order | Run Order | T_{on} | T_{off} | Current |
---|---|---|---|---|

14 | 1 | 2 | 2 | 3 |

3 | 2 | 1 | 3 | 1 |

9 | 3 | 1 | 2 | 2 |

15 | 4 | 2 | 2 | 2 |

11 | 5 | 2 | 1 | 2 |

18 | 6 | 2 | 2 | 2 |

6 | 7 | 3 | 1 | 3 |

13 | 8 | 2 | 2 | 1 |

17 | 9 | 2 | 2 | 2 |

12 | 10 | 2 | 3 | 2 |

16 | 11 | 2 | 2 | 2 |

5 | 12 | 1 | 1 | 3 |

1 | 13 | 1 | 1 | 1 |

19 | 14 | 2 | 2 | 2 |

10 | 15 | 3 | 2 | 2 |

7 | 16 | 1 | 3 | 3 |

4 | 17 | 3 | 3 | 1 |

8 | 18 | 3 | 3 | 3 |

2 | 19 | 3 | 1 | 1 |

20 | 20 | 2 | 2 | 2 |

**Table 4.**Membership function [44].

Fuzzy Number | Linguistic Scale | Fuzzy Number | ||
---|---|---|---|---|

9 | Perfect | 8 | 9 | 10 |

8 | Absolute | 7 | 8 | 9 |

7 | Very good | 6 | 7 | 8 |

6 | Fairly good | 5 | 6 | 7 |

5 | Good | 4 | 5 | 6 |

4 | Preferable | 3 | 4 | 5 |

3 | Not bad | 2 | 3 | 4 |

2 | Weak advantage | 1 | 2 | 3 |

1 | Equal | 1 | 1 | 1 |

**Table 5.**Transformation of fuzzy membership function [49].

Rank | Sub-Criteria Grade | Membership Function |
---|---|---|

Very Low (VL) | 01 | (0.00, 0.10, 0.25) |

Low (L) | 02 | (0.15, 0.30, 0.45) |

Medium (M) | 03 | (0.35, 0.50, 0.65) |

High (H) | 04 | (0.55, 0.70, 0.85) |

Very High (VH) | 05 | (0.75, 090, 1.00) |

Run Order | T_{on}(µs) | T_{off}(µs) | Current (A) | Experimental Values | Normalized Values | ||||
---|---|---|---|---|---|---|---|---|---|

V_{C}(mm/min) | MRR (mm ^{3}/min) | SR (µm) | V_{C} | MRR | SR | ||||

1 | 70 | 20 | 4 | 2.715 | 3.730 | 5.80 | 0.6632 | 0.7597 | 0.5498 |

2 | 40 | 25 | 2 | 1.234 | 1.580 | 3.27 | 0.0000 | 0.0000 | 0.0249 |

3 | 40 | 20 | 3 | 2.012 | 2.560 | 3.15 | 0.3484 | 0.3463 | 0.0000 |

4 | 70 | 20 | 3 | 2.360 | 3.000 | 5.55 | 0.5043 | 0.5018 | 0.4979 |

5 | 70 | 15 | 3 | 2.917 | 3.710 | 5.89 | 0.7537 | 0.7527 | 0.5685 |

6 | 70 | 20 | 3 | 2.441 | 3.109 | 5.20 | 0.5405 | 0.5403 | 0.4253 |

7 | 100 | 15 | 4 | 3.467 | 4.410 | 7.97 | 1.0000 | 1.0000 | 1.0000 |

8 | 70 | 20 | 2 | 1.779 | 2.260 | 4.01 | 0.2441 | 0.2403 | 0.1784 |

9 | 70 | 20 | 3 | 2.444 | 3.110 | 5.33 | 0.5419 | 0.5406 | 0.4523 |

10 | 70 | 25 | 3 | 2.033 | 2.580 | 5.22 | 0.3578 | 0.3534 | 0.4295 |

11 | 70 | 20 | 3 | 2.477 | 3.150 | 5.83 | 0.5567 | 0.5548 | 0.5560 |

12 | 40 | 15 | 4 | 3.013 | 3.830 | 3.96 | 0.7967 | 0.7951 | 0.1680 |

13 | 40 | 15 | 2 | 2.114 | 2.690 | 2.98 | 0.3941 | 0.3922 | 0.1037 |

14 | 70 | 20 | 3 | 2.486 | 3.170 | 5.00 | 0.5607 | 0.5618 | 0.3838 |

15 | 100 | 20 | 3 | 2.731 | 3.500 | 6.10 | 0.6704 | 0.6784 | 0.6120 |

16 | 40 | 25 | 4 | 1.890 | 2.400 | 4.20 | 0.2938 | 0.2898 | 0.2178 |

17 | 100 | 25 | 2 | 1.673 | 2.100 | 4.83 | 0.1966 | 0.1837 | 0.3485 |

18 | 100 | 25 | 4 | 2.490 | 3.170 | 5.70 | 0.5625 | 0.5618 | 0.5290 |

19 | 100 | 15 | 2 | 2.381 | 3.080 | 4.71 | 0.5137 | 0.5300 | 0.3237 |

20 | 70 | 20 | 3 | 2.477 | 3.210 | 5.60 | 0.5567 | 0.5760 | 0.5083 |

Source | Sum of Squares | Df | Mean Sum of Square | F Value | p-Value | Contribution | Significance |
---|---|---|---|---|---|---|---|

Model | 4.83875 | 9 | 0.53764 | 112.30 | 0.000 | 99.02% | significant |

T_{on} | 0.61454 | 1 | 0.61454 | 128.37 | 0.000 | 12.58% | significant |

T_{off} | 2.09032 | 1 | 2.09032 | 436.63 | 0.000 | 42.78% | significant |

Current | 1.93072 | 1 | 1.93072 | 403.29 | 0.000 | 39.51% | significant |

T_{on} × T_{off} | 0.01264 | 1 | 0.01264 | 2.64 | 0.135 | 0.26% | |

T_{on} × Current | 0.01514 | 1 | 0.01514 | 3.16 | 0.106 | 0.31% | |

T_{off} × Current | 0.03277 | 1 | 0.03277 | 6.84 | 0.026 | 0.67% | significant |

T_{on} × T_{on} | 0.00566 | 1 | 0.00566 | 1.18 | 0.302 | 0.12% | |

T_{off} × T_{off} | 0.00929 | 1 | 0.00929 | 1.94 | 0.194 | 0.19% | |

Current × Current | 0.07935 | 1 | 0.07935 | 16.57 | 0.002 | 1.62% | significant |

Residual | 0.04787 | 10 | 0.00479 | 0.98% | |||

Lack of Fit | 0.03694 | 5 | 0.00739 | 3.38 | 0.104 | 0.76% | Insignificant |

Pure Error | 0.01093 | 5 | 0.00219 | 0.22% | |||

Total | 4.88662 | 19 | 100.00% |

Source | Sum of Squares | Degree of Freedom | Adjusted Mean Sum of Square | F Value | p-Value | Contribution | Significance |
---|---|---|---|---|---|---|---|

Model | 8.17084 | 9 | 0.90787 | 63.96 | 0.000 | 98.29% | significant |

T_{on} | 1.02400 | 1 | 1.02400 | 72.14 | 0.000 | 12.32% | significant |

T_{off} | 3.46921 | 1 | 3.46921 | 244.39 | 0.000 | 41.73% | significant |

Current | 3.39889 | 1 | 3.39889 | 239.44 | 0.000 | 40.89% | significant |

T_{on} × T_{off} | 0.01280 | 1 | 0.01280 | 0.90 | 0.365 | 0.15% | |

T_{on} × Current | 0.02420 | 1 | 0.02420 | 1.70 | 0.221 | 0.29% | |

T_{off} × Current | 0.04205 | 1 | 0.04205 | 2.96 | 0.116 | 0.51% | |

T_{on} × T_{on} | 0.02723 | 1 | 0.02723 | 1.92 | 0.196 | 0.33% | |

T_{off} × T_{off} | 0.00066 | 1 | 0.00066 | 0.05 | 0.834 | 0.01% | |

Current × Current | 0.04975 | 1 | 0.04975 | 3.50 | 0.091 | 0.60% | |

Residual | 0.14195 | 10 | 0.01420 | 1.71% | |||

Lack of Fit | 0.11597 | 5 | 0.02319 | 4.46 | 0.0626 | 1.40% | Insignificant |

Pure Error | 0.02598 | 5 | 0.00520 | 0.31% | |||

Total | 8.31279 | 19 | 100.00% |

Source | Sum of Squares | Degree of Freedom | Mean Sum of Square | F Value | p-Value | Contribution | Significance |
---|---|---|---|---|---|---|---|

Model | 22.3776 | 9 | 2.4864 | 11.67 | 0.000 | 91.31% | significant |

T_{on} | 12.2766 | 1 | 12.2766 | 57.65 | 0.000 | 50.09% | significant |

T_{off} | 0.8762 | 1 | 0.8762 | 4.11 | 0.070 | 3.58% | Not significant |

Current | 5.1266 | 1 | 5.1266 | 24.07 | 0.001 | 20.92% | significant |

T_{on} × T_{off} | 0.5050 | 1 | 0.5050 | 2.37 | 0.155 | 2.06% | |

T_{on} × Current | 1.0440 | 1 | 1.0440 | 4.90 | 0.051 | 4.26% | |

T_{off} × Current | 0.3916 | 1 | 0.3916 | 1.84 | 0.205 | 1.60% | significant |

T_{on} × T_{on} | 0.9825 | 1 | 0.9825 | 4.61 | 0.057 | 4.01% | |

T_{off} × T_{off} | 0.3036 | 1 | 0.3036 | 1.43 | 0.260 | 1.24% | |

Current × Current | 0.2776 | 1 | 0.2776 | 1.30 | 0.280 | 1.13% | significant |

Residual | 2.1297 | 10 | 0.2130 | 8.69% | |||

Lack of Fit | 1.6794 | 5 | 0.3359 | 3.73 | 0.087 | 6.85% | Insignificant |

Pure Error | 0.4503 | 5 | 0.0901 | 1.84% | |||

Total | 24.5073 | 19 | 100.00% |

Response | Unit | Standard Deviation | R-sq | R-sq (adj) |
---|---|---|---|---|

V_{C} | mm/min | 0.0691912 | 99.02% | 98.14% |

MRR | mm^{3}/min | 0.119144 | 98.29% | 96.76% |

SR | µm | 0.461485 | 91.31% | 83.49% |

Response | Unit | Optimum Parameter Setting Considering Single Objective Optimization |
---|---|---|

V_{C} | mm/min | A_{3}B_{1}C_{3} |

MRR | mm^{3}/min | A_{3}B_{1}C_{3} |

SR | µm | A_{1}B_{3}C_{1} |

V_{C} | MRR | SR | |
---|---|---|---|

V_{C} | 1 | 1/3 | 1/7 |

MRR | 3 | 1 | 1/4 |

SR | 7 | 4 | 1 |

V_{C} | MRR | SR | |
---|---|---|---|

V_{C} | (1, 1, 1) | (0.25, 0.33, 0.50) | (0.13, 0.14, 0.17) |

MRR | (2, 3, 4) | (1, 1, 1) | (0.2, 0.25, 0.33) |

SR | (6, 7, 8) | (3, 4, 5) | (1, 1, 1) |

Weights | |
---|---|

V_{C} | (0.5286, 0.7049, 0.9312) |

MRR | (0.1486, 0.2109, 0.2996) |

SR | (0.0635, 0.0841, 0.1189) |

Alternatives | V_{C} (mm/min) | MRR (mm^{3}/min) | SR (µm) |
---|---|---|---|

1 | (0.55, 0.70, 0.85) | (0.55, 0.70, 0.85) | (0.35, 0.50, 0.65) |

2 | (0.00, 0.10, 0.25) | (0.00, 0.10, 0.25) | (0.00, 0.10, 0.25) |

3 | (0.15, 0.30, 0.45) | (0.15, 0.30, 0.45) | (0.00, 0.10, 0.25) |

4 | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) |

5 | (0.55, 0.70, 0.85) | (0.55, 0.70, 0.85) | (0.35, 0.50, 0.65) |

6 | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) |

7 | (0.75, 0.90, 1.0) | (0.75, 0.90, 1.00) | (0.75, 0.90, 1.00) |

8 | (0.15, 0.30, 0.45) | (0.15, 0.30, 0.45) | (0.00, 0.10, 0.25) |

9 | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) |

10 | (0.15, 0.30, 0.45) | (0.15, 0.30, 0.45) | (0.35, 0.50, 0.65) |

11 | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) |

12 | (0.55, 0.70, 0.85) | (0.55, 0.70, 00.85) | (0.00, 0.10, 0.25) |

13 | (0.15, 0.30, 0.45) | (0.15, 0.30, 0.45) | (0.00, 0.10, 0.25) |

14 | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) | (0.15, 0.30, 0.45) |

15 | (0.55, 0.70, 0.85) | (0.55, 0.70, 0.85) | (0.55, 0.70, 0.85) |

16 | (0.15, 0.30, 0.45) | (0.15, 0.30, 0.45) | (0.15, 0.30, 0.45) |

17 | (0.00, 0.10, 0.25) | (0.00, 0.10, 0.25) | (0.15, 0.30, 0.45) |

18 | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) |

19 | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) | (0.15, 0.30, 0.45) |

20 | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) | (0.35, 0.50, 0.65) |

Alternatives | V_{C} (mm/min) | MRR (mm^{3}/min) | SR (µm) |
---|---|---|---|

1 | (0.034, 0.059, 0.101) | (0.082, 0.148, 0.255) | (0.185, 0.352, 0.605) |

2 | (0.000, 0.008, 0.030) | (0.000, 0.021, 0.075) | (0.000, 0.070, 0.233) |

3 | (0.010, 0.025, 0.054) | (0.022, 0.063, 0.135) | (0.000, 0.070, 0.233) |

4 | (0.022, 0.042, 0.077) | (0.052, 0.105, 0.195) | (0.185, 0.352, 0.605) |

5 | (0.035, 0.059, 0.101) | (0.082, 0.148, 0.255) | (0.185, 0.352, 0.605) |

6 | (0.022, 0.042, 0.077) | (0.052, 0.105, 0.195) | (0.185, 0.352, 0.605) |

7 | (0.048, 0.076, 0.119) | (0.111, 0.190, 0.300) | (0.396, 0.634, 0.931) |

8 | (0.010, 0.025, 0.054) | (0.022, 0.063, 0.135) | (0.000, 0.070, 0.233) |

9 | (0.022, 0.042, 0.077) | (0.052, 0.105, 0.195) | (0.185, 0.352, 0.605) |

10 | (0.010, 0.025, 0.054) | (0.022, 0.063, 0.135) | (0.185, 0.352, 0.605) |

11 | (0.022, 0.042, 0.077) | (0.052, 0.105, 0.195) | (0.185, 0.352, 0.605) |

12 | (0.035, 0.059, 0.101) | (0.082, 0.148, 0.255) | (0.000, 0.070, 0.233) |

13 | (0.010, 0.025, 0.054) | (0.022, 0.063, 0.135) | (0.000, 0.070, 0.233) |

14 | (0.022, 0.042, 0.077) | (0.052, 0.105, 0.195) | (0.079, 0.211, 0.419) |

15 | (0.035, 0.059, 0.101) | (0.022, 0.063, 0.134) | (0.291, 0.493, 0.792) |

16 | (0.010, 0.025, 0.054) | (0.022, 0.063, 0.135) | (0.079, 0.211, 0.419) |

17 | (0.000, 0.008, 0.030) | (0.000, 0.021, 0.075) | (0.079, 0.211, 0.419) |

18 | (0.022, 0.042, 0.077) | (0.052, 0.105, 0.195) | (0.185, 0.352, 0.605) |

19 | (0.022, 0.042, 0.077) | (0.052, 0.105, 0.195) | (0.079, 0.211, 0.419) |

20 | (0.022, 0.042, 0.077) | (0.052, 0.105, 0.195) | (0.185, 0.352, 0.605) |

Alternatives | D+ | D– | CC_{i} |
---|---|---|---|

1 | 2.195 | 0.890 | 0.456 |

2 | 2.096 | 0.967 | 0.477 |

3 | 2.039 | 1.026 | 0.490 |

4 | 2.256 | 0.827 | 0.443 |

5 | 2.195 | 0.890 | 0.456 |

6 | 2.256 | 0.827 | 0.443 |

7 | 2.413 | 0.710 | 0.414 |

8 | 2.039 | 1.026 | 0.490 |

9 | 2.256 | 0.827 | 0.443 |

10 | 2.317 | 0.764 | 0.432 |

11 | 2.256 | 0.827 | 0.443 |

12 | 1.918 | 1.151 | 0.522 |

13 | 2.039 | 1.026 | 0.490 |

14 | 2.112 | 0.960 | 0.473 |

15 | 2.341 | 0.764 | 0.427 |

16 | 2.173 | 0.898 | 0.460 |

17 | 2.231 | 0.839 | 0.448 |

18 | 2.256 | 0.827 | 0.443 |

19 | 2.112 | 0.960 | 0.473 |

20 | 2.256 | 0.827 | 0.443 |

Performance Response | Optimal Setting | Predicted Values | Experimental Values | % Error |
---|---|---|---|---|

V_{C} (mm/min) | T_{on} 40 µs, T_{off} 15 µs, Current 2A | 2.067 | 2.114 | 2.22 |

MRR (mm^{3}/min) | 2.616 | 2.690 | 2.75 | |

SR (µm) | 3.117 | 2.98 | 4.39 |

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**MDPI and ACS Style**

Fuse, K.; Dalsaniya, A.; Modi, D.; Vora, J.; Pimenov, D.Y.; Giasin, K.; Prajapati, P.; Chaudhari, R.; Wojciechowski, S.
Integration of Fuzzy AHP and Fuzzy TOPSIS Methods for Wire Electric Discharge Machining of Titanium (Ti6Al4V) Alloy Using RSM. *Materials* **2021**, *14*, 7408.
https://doi.org/10.3390/ma14237408

**AMA Style**

Fuse K, Dalsaniya A, Modi D, Vora J, Pimenov DY, Giasin K, Prajapati P, Chaudhari R, Wojciechowski S.
Integration of Fuzzy AHP and Fuzzy TOPSIS Methods for Wire Electric Discharge Machining of Titanium (Ti6Al4V) Alloy Using RSM. *Materials*. 2021; 14(23):7408.
https://doi.org/10.3390/ma14237408

**Chicago/Turabian Style**

Fuse, Kishan, Arrown Dalsaniya, Dhananj Modi, Jay Vora, Danil Yurievich Pimenov, Khaled Giasin, Parth Prajapati, Rakesh Chaudhari, and Szymon Wojciechowski.
2021. "Integration of Fuzzy AHP and Fuzzy TOPSIS Methods for Wire Electric Discharge Machining of Titanium (Ti6Al4V) Alloy Using RSM" *Materials* 14, no. 23: 7408.
https://doi.org/10.3390/ma14237408