Development of AHP-embedded Deng’s hybrid MCDM model in micro-EDM using carbon-coated electrode

2.1 Experimental setup

The carbon-coated WC electrode was used in this study, and the coating thickness of the electrode is approximately 10 μm. The results of EDX analysis, shown in Figure 1, show that the thin film coating consists of elements W, O, C, Co, and Zn. The chemical vapor deposition (CVD) was used for coating on tungsten carbide substrate up to 10 μm. The surface quality of the coated electrode is quite good (Figure 2). The structural feature consists of small particles, and the sizes of the particles are quite similar (Figure 3). The micro-EDM machine used in this experimental study is the Hyper 10 micro-EDM. The dielectric fluid is EDM oil, and the workpiece material is Ti-6Al-4V. Quality indicators, including Z and TWR, were used in this study. Process parameters are investigated in micro-EDM as given in Table 1.

Figure 1

EDX coating of electrode.

Figure 2

Photograph of electrode surface.

Figure 3

Grain structure in the coating.

2.2 Methodology

Taguchi–Deng’s method was used for multi-objective decision making in this study. The steps to combine Taguchi with Deng’s are shown in Figure 4. The optimal value should be determined by S/N factor analysis, and the value of S/N is determined as shown in (1)–(3).

Figure 4

Steps of Taguchi–Deng’s method.

* The S/N ratio:

The “larger the better” S/N value is closer to optimal parameters, and the results of the indicators are as follows:

where mean square deviation (MSD_HB) can be defined by the following expression:

In this case, r represents the number of repetitions, and y _i is the value of experiment results.

Similarly, “Normal, the best” S/N ratio can be defined as follows:

where the mean square deviation for nominal the best (MSD_NB) can be defined as:

In this expression, y ₀ represents the standard values or target values.

The “Lower the better” S/N ratio can be defined as:

where mean square deviation for lower the better (MSD_LB) can be defined as:

In this equation, ∑ i = 1 r y i 2 is the overall typical value of each experiment.

*Multi-response optimization methodology by Deng’s Similarity-Based Method:

Hepu Deng (2007) proposed a new approach to find the best alternative solution to a multi-criteria decision-making process [19]. This approach compares the magnitude and conflict between the alternative and ideal solutions. Gradients of variables indicate conflicts, and the best choice can be identified with the rank of conflicting indexes. The calculation and ranking process in AHP–Deng’s is described in previous study [20].

Step 1: Sort the selected indicators in the matrix form:

where x _ij is the experimental value of the selected evaluation indicators (j is the number of indicators (j = 1 – n), and i is the number of experimental values of an indicator (i = 1 – m)).

Step 2: Normalize the determination matrix and convert the responses to non-dimensional form to make comparisons between response values. The standardized matrix is established through standardized values x ij ′ ( 0 ≤ x ij ′ ≤ 1 ) . Determine the normalized determination matrix with the normalized value x ij ′ as (6).

Step 3: Calculate the performance matrix by assigning the weight of quality indicators to the normalized matrix. The quality matrix is defined as follows:

where A _ij is the value of quality indicator j in the ith experiment, W _j is the weightage of the indicators, and A is the quality matrix.

Step 4: Positive ideal solution (PIS) and negative ideal solution (NIS) are determined as follows:

PIS:

NIS:

where y j + = { max j = 1 , 2 , 3 , … m , y m ′ } is the best value of x _j , y j − = { min j = 1 , 2 , 3 , … m , y j ′ } is the worst value of x _j and A i = { y 1 ′ , y 2 ′ , … , y n ′ } , J is combined with good indicators, and J′ is combined with bad indicators.

Step 5: Calculate the degree of conflict between each alternative: The degree of conflict between each alternative solution and PIS and the NIS can be calculated as follows:

The conflict between the alternative solution and PIS can be obtained as:

The conflict between the alternative solution and NIS can be obtained as:

Step 6: The degree of similarity and conflict between the alternative solution and the PIS and NIS are calculated as follows:

A degree of conflict:

A degree of similarity:

Step 7: Calculate the overall performance index:

The overall performance index for each alternative is calculated as follows:

The larger the index value is, the more priority it will have.

Step 8: Rank according to Deng’s similarity-based method. Higher P _i values will give good-quality Ai solutions.

3.1 Determination of the experiment with the best results by Deng’s method

The results are described in detail as follows:

Step 1: Criteria’s matrix:

Step 2: Normalize the criteria’s matrix (Table 2).

Step 3: Criteria’s weights: The value of the weights of the quality criteria includes W _Z = 0.667 and W _TWR = 0.333. The results are shown in Table 3.

Step 4: The best solution (A ⁺) and the worst solution (A ⁻): the result is as follows: A ⁺ = {Z = 0.325; TWR = 0.17} and A ⁻ = {Z = 0.089; TWR = 0.196}.

Step 5: cos θ i + and cos θ i − : The results are described in Table 3.

Step 6: DC⁺, DC^-, DS⁺, and DS⁻ are determined as in Table 3.

Step 7: P _i are determined as in Table 3.

Step 8: Ranking: The value of P _i is the largest with Exp. No. 1. The best result of the multi-objective problem includes process parameters (V = 120 V, C = 100 pF, and RPM = 200 rpm), and the value of quality indicators (Z = 0.652 mm and TWR = 0.010 µm).

3.2 ANOVA of P _i in Deng’s method

The larger the value of P _i , it will lead to an increase in the efficiency of the multi-objective optimization problem. The changes of the levels of the parameters V, RPM, and C were different, which resulted in P _i being changed very differently (Figure 5). P _i is maximum at V = 120 V, C = 100 pF, and RPM = 200 rpm/min, and it is the smallest with V = 140 V, C = 1,000 pF, and RPM = 400 rpm/min. The rank of influence of process parameters on P _i is shown in Table 4. The results show that C has the most significant influence on P _i , and V has the least influence on Pi. The influence of interaction pairs between process parameters on P _i is also shown in Figure 6. The results show that the pair of interactions between RPM and V will produce the largest number of P _i ≥ 0.052 (Figure 8a). This result proves that this pair of interactions will give the best performance for this multi-objective decision problem.

Figure 5

Influence of process parameters on P _i in Deng’s method.

Figure 6

Contour plots for P _i in Deng’s method.

3.3 Determination of optimal results by analysis of S/N

Optimal process parameters of Deng’s method are shown in Figure 8, and the result is V = 120 V, C = 100 pF, and RPM = 200 rpm. The results of the quality criteria under optimal conditions are calculated by (15) (Table 5). Compared with the calculated results of the quality criteria, the experimental results of Z and TWR have good accuracy (Figure 7).

Figure 7

Analysis of S/N of Deng’s.

3.4 Machining quality under optimum conditions

The SEM figure of the machined hole showed that the quality of the hole machined under optimum conditions was significantly improved (Figure 8). Smaller particles adhere to the surface of the machined hole under optimal conditions. This can contribute to improved surface finish after micro-EDM with optimal conditions. The size of the machined hole under optimal conditions is 19.05% smaller than it was in Exp. No. 9. This will affect the machining accuracy in micro-EDM. The machined surface is adhered to by many particles as shown in Figure 9. Their sizes are very different and the particles are randomly distributed over the machined surface. The reason is that the workpiece and electrode materials are melted and evaporated by the thermal energy of the electric sparks, and they are immediately affected by the dielectric solution. Therefore, they firmly adhere to the machined surface. Figure 10 also shows that voids and micro-cracks are also present in significant quantities on the machined surface. These have significantly affected the working quality of the machined surface after micro-EDM with the coated electrode.

Figure 8

SEM ×100 of machining hole. (a) Exp. No. 9 (the worst optimal effect) and (b) Exp. No. 1 (the best optimal effect).

Figure 9

Topography of machined surface.

Figure 10

Cracks on machined surface.