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Abstract

Selection of optimal processing condition is an issue associate with multi-criteria decision making (MCDM) which can be influenced by several conflicting processing variables. The present study have been devoted to the implementation of TOPSIS algorithm in coordination with the VIKOR technique to yield an optimistic processing criteria while machining of metal matrix composites under electro-discharge machining process. A decision-making module has been developed for selection of optimum processing conditions under specified machining conditions. An experimental investigation was performed based on Taguchi’s orthogonal array on the newly prepared AA6061-cenosphere MMCs to analyze the sensitivity of EDM attributes to the process parameters such as peak current, pulse on time, the percentage of reinforcement, and flushing pressure. The weighing factors for the criteria were determined using AHP method. The results concluded that both the applied MCDM approaches, TOPSIS and VIKOR accrue the similar possible optimal solution having optimal value of pulse current 10 A, pulse on time 1010 µs, percentage of reinforcement as 2% and flushing pressure as 0.6 MPa.

Keywords

EDM TOPSIS VIKOR MCDM AMCs

Introduction

With ever increasing economic demand for high strength structural materials having high strength to weight ratio in modern manufacturing is influenced by the advancements in the field of manufacturing research. Aluminum based metal matrix composites (AMCs) have gathering more attention in manufacturing industries due to their excellent strength, higher resistance to wear, high-temperature resistance, and stiffness.^1–3 The major hurdle in the application of the existing AMCs is its expensive fabrication process which can be minimized by incorporating cost-effective waste by-product of coal combustion such as natural minerals and Fly ash.^4–7 Proper incorporation and homogeneous distribution of the ceramic particles within the slurry and also acquiring appropriate interfacial interaction between them would often requires to the improvement of tribo-mechanical properties.^8–10 Hence semi solid casting processes like stir casting and modified stir casting (also known as compocasting) are the possible alternations.^11–19 Although, the applications of MMCs are limited where as their machinability remains a serious concern. Difficulties was arises during machining using conventional techniques. Hence, the non-conventional processing are the possible alternatives in terms of qualitative and economic view point.^20,21 Keeping view in growing economic demands in the manufacturing industries, processing of quality components leads to enhancing the overall cost of production. In such scenario, selection of optimal processing condition for machining of such difficult of machined materials become most vital. Therefore, it is becoming an essential measure to select an optimal machining condition. In EDM, several factors influencing the quality of machining process. These factors in turn, provide the criteria for the evaluation of the finished products.

Prakash et al.²² made an advanced review on conventional and micro–EDM of difficult-to-machine materials such as nickel and its alloys, titanium alloys, stainless steel (SUS 304), and advanced Metal matrix composites and ceramics. The review discusses the current research trends and developments, research gaps, and challenges of the conventional electrical discharge machining of those difficult to machined materials particularly of MMC’s in details. Gohil and Puri²³ investigated the effects of crucial machining parameters on material removal rate (MRR) and surface roughness (Ra) during electro discharge turning operations of titanium alloy Ti-6Al-4V. Taguchi’s orthogonal array had been implemented to design the experiments. The signal-to-noise ratio analysis is employed to find the optimal operating condition. The experimental results concluded that peak current, gap voltage, and pulse-on time are the most influencing parameters that contribute more than 90% to MRR. In the context of Ra, peak current, and pulse-on time come up with more than 82% of contribution. Rouniyar and Shandilya²⁴ studied the effects of individual machining parameters on Al6061 alloy during machining on magnetic field assisted powder mixed electrical discharge machining setup. The effect of peak current was observed to be dominant on material removal rate and tool wear rate followed by pulse on time, powder concentration and magnetic field. Increase in MRR and TWR was observed with increase in peak current, pulse on time, and a decrease in pulse off time, whereas, for MRR increases and TWR decreases up to the certain value and follow the reverse trend with an increase in powder concentration. Pramanik et al.²⁵ reported the machinability of 10 vol% of SiC reinforced Al6061 alloys using wire EDM. It was observed that the reinforced particles size, wire tension, and pulse-on time has influence significantly on the diameter error, circularity, and surface roughness of the composites respectively. It has been reported that there are more surface defects encountered when particle sizes are smaller, and circularity is improved when particles are in a medium size. In addition, the surface defect gets reduced as the particles increase the melting resistance of the surface. These criteria are generally MRR, EWR, SR, Process Time, Process Energy, Dielectric Consumption, etc. As in a group these criteria’s are conflicting in nature like more MRR is required for good machining which results higher roughness in the generated surface and vice-versa. These conflicting nature of machining response leads to presume weightage based criteria selection for optimum results. The operator has to have the flexibility to provide diversified weightage among multiple conflicting machining responses based on requirement for industrial applications. Therefore, selection of optimal processing condition in a sophisticated process in EDM, and therefore it is necessity to incorporate Multi-criteria Decision Making approaches. MCDM is a sub-discipline of operational research, which can be used to evaluate conflicting criteria in decision making. Among the various MCDM techniques, TOPSIS has been found to applied in a broader scale for optimal parameter selection of Electrical Discharge Machining because of its ability to handle both continuous and discrete data.^26–28 The application of TOPSIS is also not limited to machining environment but also stock exchange too.^29,30 Similarly, VIKOR has been used extensively for selection of machining environment. VIKOR is also applied for selection of lean tool to obtain the accuracy of transmission system and thereby allocation for multi-axis machine tools.^31–34

Most of the findings reported were confined to particular techniques of either TOPSIS or VIKOR separately. Although VIKOR in conjunction with TOPSIS has been used as effect alternations for estimation of wall insulations, finding the ranking of redevelopment of buildings and selection for supplier of construction projects.^35–37 The implementation of TOPSIS in conjunction with VIKOR approach for finding the optimal processing conditions while processing through electric discharge machine of such adverse characteristics of AMC’s are in particular and not reported any related finding till date. Although, in some earlier publications of the authors has predicted the optimal machining conditions under specified domain followed by CCD of RSM technique, but there also found few constrains which barricades the implementation of those techniques.^{38–40,44,50,51} Thus, TOPSIS in combined with VIKOR MCDM can well predict the influence of process variables on performance characteristics and hence, is a proper approach for finding the optimal processing conditions. In the present study, the TOPSIS and VIKOR method are employed for finding the optimal parametric condition while machining the newly prepared AMCs through EDM.

Materials and methods

The preheated (<280°C) fly ash cenosphere particles with measured quantity was incorporated into the coated graphite crucible containing AA6061 aluminum alloy at a furnace temperature of approx. 610°C. Simultaneous stirring was done so as to obtain homogeneous distribution of the particles into the matrix. Tables 1 and 2 demonstrate the chemical composition of the aluminum alloy and cenosphere respectively. The average particles sizes of the cenospheres were considered as 45–50 µm and density was 2300 kg/m³. The prepared meals were then used for the preparation of test specimen after stirring. Temperature difference has been maintained between the pouring and the casting temperature so as to obtain a better fluidity of the composite slurry. Composites were fabricated for different percentage of (0%, 2%, 4%, 6%, and 10%) cenosphere particles. The compo casting setup is shown in Figure 1.

Table 1.

The chemical composition of AA6061 Alloys.

Element	Mg	Si	Fe	Mn	Cu	Cr	Zn	Ni	Ti	Other
Wt.%	0.96	0.52	0.24	0.11	0.13	0.08	0.06	0.02	0.01	Balance

Table 2.

Chemical composition and the weight percentage of cenosphere fly ash.

Element	SiO₂	Al₂O₃	Fe₂O₃	MgO	TiO₂	K₂O	MnO	others
Wt.%	52.9	25.51	8.93	1.8	2.39	0.9	0.6	Balance

Figure 1.

Ideal and compromise solutions.

Experimental details

The current study associated with the statistical techniques implementing for modeling and optimizing the performance measures of the machining called as Response Surface Method (RSM), which correlates the quantitative processing parameters⁴⁸. A quadratic polynomial model has been derived in order to explore the performances of the independent system variables toward response characteristics as also called the regression model as shown in equation (1). The regression coefficients of the predicted model can be approximated by the experimental data analysis following the statistical software package, “Design Expert 10.0.”

Y = C_{0} + \sum_{i = 1}^{n} C_{i} X_{n} + \sum_{i = 1}^{n} d_{i} X_{i}^{2} \pm ε

(1)

Where, Y is the estimated response, C’s are the coefficients and x_i’s are the independent variables. This assumed surface Y contains linear, squared, and cross product terms of variables x_i’s, a number of experimental design techniques are available in order to estimate the regression coefficient.

Central composites design (CCD) of RSM has been implemented to design the experimentation in order to reduce the total number of experimental trails. All combinations of factors at two levels (high, +1 and low, −1) of full factorial CCD that comprises of eight-star points, and six central point’s (coded level 0), corresponds to an α value of 1 has been considered. Typical pictorial representations of face centered CCD element have been shown in the diagram below. CCD is first-order (2N) designs increased by additional center and axial points to allow estimation of the tuning parameters of a second-order model. Figure 2 shows a CCD for three design variables. In Figure 2, the design involves 2N factorial points, 2N axial points and one central point. The factorial points contribute to the estimation of linear terms and two factor interaction terms. Contribution of interactions terms can be only estimated by the factorial points. The axial points contribute to the estimations of quadratic terms. If the axial points are not found, than only the sum of quadratic terms can be estimated. The central run provides an internal estimate of pure error and contributes toward the estimation of quadratic terms. CCD presents an alternative to 3N designs in the construction of second-order models as the number of experiments are reduced as compared to a full factorial design (15 in the case of CCD compared to 27 for a full-factorial design). The number of factorial run depends on the types of factorial design used and the number of factors. The experiments may be time consuming even with the use of CCD, in case of problems with a large number of designs variables.

Figure 2.

Representation of CCD for 3 design variables.

A total of 30 experimental trails have been conducted based on experimental design with a combination of four distinct process parameters in this study. The process variables and their operating domain with actual and coded factors have been revealed in Table 3. The results obtain after conducting the experimental runs have been shown in Table 4.

Table 3.

EDM Machining parameters with their levels.

Parameters	Labels	Levels
		−1	0	+1
Pulse current, Ip (A)	A	6	8	10
Pulse on time, T_on (µs)	B	210	604	1010
Percentage of reinforcement, vol%	C	2	4	6
Flushing pressure, P_flu (MPa)	D	0.2	0.4	0.6

Table 4.

Design layout and experimental results.

Exp. no.	Factor 1 A: I_p	Factor 2 B: T_on	Factor 3 C: vol%	Factor 4 D: P_flu	Response 1 MRR (g/min)	Response 2 EWR (g/min)	Response 3 SR (µm)
1	6	210	2	0.2	0.0999	0.0028	7.7919
2	10	210	2	0.2	0.1619	0.0041	10.5094
3	6	1010	2	0.2	0.4156	0.0013	10.7293
4	10	1010	2	0.2	0.6248	0.0028	13.8752
5	6	210	6	0.2	0.0751	0.0032	8.9826
6	10	210	6	0.2	0.1698	0.0192	12.8739
7	6	1010	6	0.2	0.3856	0.0029	12.9837
8	10	1010	6	0.2	0.4867	0.0108	16.7853
9	6	210	2	0.6	0.1006	0.0047	7.8934
10	10	210	2	0.6	0.1721	0.0051	11.1073
11	6	1010	2	0.6	0.3554	0.0015	10.9973
12	10	1010	2	0.6	0.6843	0.0019	14.2872
13	6	210	6	0.6	0.0887	0.0034	9.0183
14	10	210	6	0.6	0.1997	0.012	12.8641
15	6	1010	6	0.6	0.3891	0.0025	13.4281
16	10	1010	6	0.6	0.4998	0.0028	16.8724
17	8	604	4	0.4	0.1057	0.001	10.9279
18	8	604	4	0.4	0.6248	0.0028	13.8752
19	8	604	4	0.4	0.4867	0.0108	16.7853
20	8	604	4	0.4	0.1049	0.0016	10.6884
21	6	604	4	0.4	0.0875	0.0009	9.1284
22	10	604	4	0.4	0.1893	0.0037	11.8073
23	8	210	4	0.4	0.0985	0.0061	8.829
24	8	1010	4	0.4	0.3928	0.0012	11.9719
25	8	604	2	0.4	0.1142	0.0017	9.2943
26	8	604	6	0.4	0.0929	0.0023	11.0981
27	8	604	4	0.2	0.1011	0.0032	10.6172
28	8	604	4	0.6	0.1092	0.0014	10.9891
29	8	604	4	0.4	0.1049	0.0021	10.6503
30	8	604	4	0.4	0.1042	0.0033	10.5173

Performing experiments

Die-sinking EDM machine were used to carry out the experiments for time duration of 20 min so as to minimize the redundancy of errors as tabulated in Table 4. The work piece materials of size 20 mm × 30 mm and 5 mm thick and 10 mm diameter of electrolytic copper electrode was served as EDM tool. Circular electrode has been preferred to accumulate higher MRR and lower TWR. The spark eroded materials and the debris particles from the machining zone has been removed by using impulse jet flushing system technique with hydrocarbon oil as the dielectric fluid. The MRR and TWR values have been measured by the weight difference method employing a digital weight balance of 0.001 g precision. Figure 3 have been delineated the schematic of the machining process.

Figure 3.

Plot of $S_{i,} R_{i}, Q_{i} and R C_{i}$ .

Parametric analysis using TOPSIS and VIKOR

The Technique for Order of Preference for Similarity to Ideal Solution also known as TOPSIS is one of the most extensively used method for solving MCDM problems.^41–43 It compares the alternatives solution by measuring the ideal and anti-ideal solutions at a distance from a positive ideal solution. To generate maximum profit and to minimize the cost criteria, the decision maker has preferred the positive ideal solution and the negative ideal solution has been preferred by the decision maker to minimize the profit criteria and to maximize the cost criteria. Hence, the best alternative choice is closest to the positive ideal solution and furthest from the negative ideal solution too.

The following steps are involved in TOPSIS method:

Step 1: Determination of decision matrix “X.”

It has been consider that there are “n” numbers of possible alternatives decision-makers, called $A = {A_{1}, A_{2} A_{3} A_{4}, \dots \dots . A_{n}}$ , which can be estimated against “m” criteria $C = {C_{1}, C_{2} C_{3} C_{4}, \dots \dots . C_{n}}$ . A performance ratings has been evaluated by the decision matrix $D_{m \times n}$ for each alternative concerning criteria for every decision maker, which can be identified by $x_{ij}$ where the “i” and “j” represents the alternative and the criteria respectively. Hence the decision maker matrix can be expressed as

\begin{matrix} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ . \dots . \\ A_{2} \end{matrix} | \begin{matrix} \begin{matrix} C_{1} & C_{2} C_{3} C_{4} \dots \dots \dots \dots \dots \dots \cdot\cdot C_{m} \\ x_{11} & x_{12} x_{13} x_{14} \dots \dots \dots \dots \dots \dots \cdot\cdot x_{1 m} \end{matrix} \\ x_{21} x_{22} x_{23} x_{24} \dots \dots \dots \dots \dots \dots \cdot\cdot x_{2 m} \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ x_{n 1} x_{n 2} x_{n 3} x_{n 4} \dots \dots \dots \dots \dots \dots \cdot\cdot x_{n m} \end{matrix} |

Step 2: Evaluation for the normalized decision matrix

The decision matrix $D_{m \times n}$ is then converted into the normalized decision matrix R is calculated considering vector normalization, that is

R = {(r_{ij})}_{m \times n}

(2)

Where, $r_{ij} = \frac{x_{ij}}{\sqrt{\sum_{i = 1}^{n} x_{ij}^{2}}}$ i = 1,2,3,…n; j = 1,2,3,…m;

Step 3: Calculation of the weighted normalized decision matrix

Different weight age have been assigned to the evaluation criteria, depending upon the requirement of the decision makers, Hence, it is required to evaluate the weighted normalized decision matrix.

The criteria weights to be to be assumed as $w_{j} (j = 1, 2, 3, \dots m;)$

\begin{matrix} w = {(w_{ij})}_{m \times n} = {(w_{j} \times r_{ij})}_{m \times n,} \\ i = 1, 2, 3, \dots n; j = 1, 2, 3, \dots m; \end{matrix}

(3)

Step 4: Evaluating the positive and the negative ideal solution

\begin{matrix} PI S_{i} = {min (t_{ij,} i = 1, 2, 3, \dots n) Ij \in J ., \\ max (t_{ij,} i = 1, 2, 3, \dots n) Ij \in J + λ} \\ NI S_{i} = {min (t_{ij,} i = 1, 2, 3, \dots n) Ij \in J ., \\ max (t_{ij,} i = 1, 2, 3, \dots n) Ij \in J + λ} \end{matrix}

(4)

Step 5: Calculation of distance from each alternative to the PIS and NIS,

\begin{matrix} d_{iPIS} = \sqrt{\sum_{j = 1}^{m} {(t_{ij} - PI S_{ij})}^{2}}, i = 1, 2, 3, \dots n \\ d_{iNIS} = \sqrt{\sum_{j = 1}^{m} {(t_{ij} - NI S_{ij})}^{2}}, i = 1, 2, 3, \dots n \end{matrix}

(5)

Step 6: Evaluation of the relative proximity of each alternative to the positive ideal solution and negative ideal solution.

R C_{i} = \frac{d_{iNIS}}{d_{iPIS} + d_{iNIS}}

(6)

It must be noted here that, $R C_{i} = 1$ if and only if the ith solution is the PIS and $R C_{i} = 0$ if and only if the ith solution is the NIS.

Step 7: Ranking of alternatives

Based on the closeness relative to the ideal solution, the alternatives are ranked accordingly.

It must be noted that TOPSIS minimizes the deviation to the ideal alternatives and maximizes the deviation to the nastiest solutions.

A stepwise TOPSIS flowchart is shown in Figure 4.

Figure 4.

Step-wise TOPSIS flow chart.

The Multi criteria decision optimization for a complex system has also analyses by VIKOR. It determines compromise ranking-list from the compromise solution followed by the weight stability intervals for preference stability of the compromise solution obtained with the initial weights. Most prominently the VIKOR method has been focuses on selecting and ranking from a set of alternatives, during conflicting criteria consideration. It was developed by Serafim Opricovic, and based on the measure of closeness to the ideal solution,⁴⁹ the ranking index has been made.

The alternatives are denoted by $A = A_{1}, A_{2}, A_{3}, A_{4}, \dots \dots A_{i}$ where is “i” represents the numbers of total alternatives. For the alternative $A_{i}$ the rating of mth criteria is denoted by $F_{mi}$ .

The maximum group utility (majority rule), have been provide the solution for $mi n_{i} i_{i}$ , and the solution obtained by $mi n_{i} R_{i}$ with a minimum individual regret of the opponent.⁴⁰

The compromise solution, also known as $F^{compromise}$ , is a feasible solution (closest to the ideal solution) that is $F^{ideal}$ , and depending on the mutual concessions, an agreement has been established so called compromise which is delineated in Figure 5 by $Δ F_{1} = F_{1}^{ideal} - F_{1}^{compromise}$ and $Δ F_{2} = F_{2}^{ideal} - F_{2}^{compromise}$ .

Figure 5.

Schematic diagram of die sinking EDM setup.

The steps involved in VIKOR to determine the comprise solution are as follows:

(a) Determination of the $F_{j}^{+}$ (best) and the $F_{j}^{-}$ (worst) of all criteria functions, that is $j = 1, 2, 3, 4, \dots . . m$ where, j denoted the benefit criteria.

F_{j}^{+} = mi n_{i} F_{mi,} F_{j}^{-} = ma x_{i} F_{mi,}

(7)

Moreover, if “i” denotes cost criteria then

F_{j}^{+} = ma x_{i} F_{mi,} F_{j}^{-} = mi n_{i} F_{mi,}

(b) Computation of Utility Measure ( $S_{j}$ ) and Regret Measure ( $R_{j}), i = 1, 2, 3, 4, \dots . . n .$

S_{i} = \sum_{j = 1}^{m} w_{j} \frac{F_{j}^{+} - F_{mi}}{F_{j}^{+} - F_{j}^{-}}

(8)

And

R_{i} = ma x_{j} [w_{j} \frac{F_{j}^{+} - F_{mi}}{F_{j}^{+} - F_{j}^{-}}]

(9)

Here, $w_{i}$ , expressing the relative importance of weight age.

Q_{i} = v \frac{(S_{i} - S^{+})}{(S^{-} - S^{+})} + (1 - υ) \frac{(R_{i} - R^{+})}{(R^{-} - R^{+})}

(10)

Here, $S^{+} = mi n_{i} S_{i,} S^{-} = ma x_{i} S_{i,}$

R^{+} = mi n_{i} R_{i,} R^{-} = ma x_{i} R_{i,}

Where, $v$ denoted the strategical weightage for the utility of maximum group, generally $v = 0.5$ has been consider.

(d) The alternatives Ranking in ascending order for $S_{i,} R_{i}, Q_{i}$ values. These lists provide the best alternative according to themselves. The lower the value is the closer is the alternative to the ideal solution.

(e) A compromise solution has been proposed as an alternative ( $a'$ ) which can be ranked as $, Q_{i}$ (i.e. contains the minimum value of $Q_{i}$ subjected to satisfying the following conditions –

(i) Condition 1: adequate advantage

Q (a ″) - Q (a') \geq DQ

$Here, a^{″}$ represents the alternative with the second ranking position in the list by $Q_{i}$

$DQ = \frac{1}{J - 1}$ Where, J denotes the number of alternatives in the experiment.

(ii) Condition 2: adequate stability in decision making

For being the compromise solution, the alternative have been considered to be the best ranked by $S_{i}$ and/or $R_{i}$ . As in the decision-making process, the compromise solution must be stable, which could be –

Selection by the rule of majority: (when $ϑ$ > 0:5 is needed), or

By agreement: ( $ϑ \approx$ 0.5), or

With prohibition ( $ϑ$ < 0:5).

Here, the weight strategy is represents by $ϑ$ for the maximum group utility.

Violations of any of these conditions, results in prose compromise solutions which composed as:

If, Condition 2 is not satisfied, the alternatives are $a'$ and $a ″$ ,

If, Condition 1 is not satisfied, the alternatives $a', a ″, a'^{″}, \dots \dots a^{(N)}$ ; and $a^{(N)}$ can be estimated by the relation $Q (a^{N}) - Q (a') < DQ$ for maximum N (considering the alternatives positions are within the range).

A stepwise VIKOR process applied here is exposed in Figure 6.

Figure 6.

A stepwise VIKOR process.

Weight calculation through AHP

Both the methods, TOPSIS and VIKOR, use criteria weights to calculate the optimal alternatives. There are several techniques to evaluate the weight criteria. In the present study, the Analytical Hierarchical Process have been implemented for the evaluation of the weight criteria. The AHP method has been used extensively in solving the multiple criteria decision-making problems.⁴⁵ The purpose of AHP has been capturing the expert’s knowledge in the field of application. The AHP method helps user to derive priorities among criteria and alternatives, thereby providing measures of judgment consistency of the expert. The AHP process was based on pair wise evaluation criteria.

To find the weights for the criteria by AHP following steps are involved –

(a) Developing a pair wise evaluation matrix for each criterion.

The pair wise evaluation matrix G ia a $m \times n$ matrix, where “m” represents the number of evaluation criteria. In the matrix G, each entry $g_{m' m}$ represents the importance of mth criterion concerning the m′th criterion. It $g_{m' m} > 1$ , then m′th criterion is more critical than mth criterion; and vice versa. If there are same significance in two criteria, then the entry $g_{m' m}$ is 1. It must be noted that for entries $g_{m' m}$ and $g_{m m^{'}}$ , there exists a relation, that is

g_{m' m} \times g_{m m^{'}} = 1

The relative significance between the two criteria is evaluated based on the numerical scale of 1 to 9 as depicted in Table 1.⁴⁶

In AHP method the number of evaluation criteria should never exceed nine, it because of following reasons –

Following the study published by George A Miller with the limits of capacity for human toward information processing.⁴⁷

As per Saaty and Ozdemir, the number of pair wise comparisons increases with the number of criteria, it is $(n^{2} - n) / 2 .$ For nine criteria the number of comparisons will be 36.⁴³ For a higher number of comparisons, logical inconsistency may arise, and the consistency ratio may exceed the value of 0.1. Thus making the AHP results questionable.

An essential pairwise comparison matrix is shown below –

\begin{matrix} C_{1}^{'} \\ C_{2}^{'} \\ C_{3}^{'} \\ C_{4}^{'} \\ \dots . . \\ C_{m}^{'} \end{matrix} | \begin{matrix} \begin{matrix} C_{1} & C_{2} \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots C_{m} \\ 1 & g_{12} \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots g_{1 m} \end{matrix} \\ g_{21} 1 . . \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots . . g_{1 m} \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \\ g_{m 1} g_{m 2} \dots \dots \dots \dots \dots . \dots \dots \dots \dots \dots \dots \dots 1 \end{matrix} |

Here, $C_{1}^{'}$ is actually $C_{1};$ different representations are used to avoid confusion.

(b) Normalization of matrix G

The normalized pairwise comparison matrix is derived from the matrix G by putting one in a column containing total sum of the entries. It means now the following expression calculates each entry of the normalized matrix.

{\bar{g}}_{m' m} = \frac{g_{m' m}}{\sum_{i = 1}^{m'} g_{lm}}

(11)

The criteria weight vector was calculated from the average values of the entries on each row of the normalized pairwise comparison matrix.

w_{m} = \frac{\sum_{i = 1}^{m} g_{m' m}}{m}

(12)

The consistency of the calculated weight vector can be checked by calculating the consistency ration of the calculation.

Results and discussions

In the present study, selection of an optimal alternative for the electric discharge machining through two MCDM techniques, namely called TOPSIS and VIKOR, have been implemented. TOPSIS will give a single best alternative while VIKOR may provide a compromise solution or a set of compromise solutions. Depending on the result of both these methods, one will get the optimal solution for the machining process.

Before proceeding forward, it is essential to find the criteria weight, as both TOPSIS and VIKOR method applies criteria weight to find the optimal solution. The weight of the criteria is found using AHP (as discussed earlier).

It involved following steps to find the weight of the criteria –

Step 1: Pair wise comparison matrix.

The pair wise comparison matrix for the problem is shown in Table 5.

Table 5.

Pair-wise comparison value scale.

Scale	Explanation	Value
Equal importance	Equal contribution is given by two criteria toward objective	1
Moderate importance	Experience and judgment are considered to be slightly in favor	3
Strong importance	Experience and judgment are considered to be strongly in favor	5
Very strong importance	Experience and judgment are very strongly favor, that is dominance	7
Absolute importance	The judgment favoring one activity over another assuming highest possible order of affirmation.	9
Here, 2, 4, 6, 8 will be the intermediate values in the comparison scale.

Step 2: Normalization of matrix G

The normalized pairwise comparison matrix for the problem is calculated as per the equation (11) and shown in Table 6.

Table 6.

Pair wise comparison matrix.

Factor	$C_{1}$	$C_{2}$	$C_{3}$
$C_{1}^{'}$	1.00	6.00	5.00
$C_{2}^{'}$	0.17	1.00	2.00
$C_{3}^{'}$	0.20	0.50	1.00

Step 3: Criteria weight vector

The criteria weight vector is calculated using equation (12) and shown in Table 7.

Table 7.

Criteria weight vector.

Factor	Weight
$C_{1}^{'}$	0.72
$C_{2}^{'}$	0.17
$C_{3}^{'}$	0.11

The consistency ration calculated for the above AHP calculation was found to be less than 0.1. Therefore, the judgment made of the calculation was consistent.

After calculation of the weight vector criteria, the optimal value of machining through TOPSIS and VIKOR was calculated. As both the methods use decision matrix for calculation therefore while experimenting a group of four decision-makers namely Pulse Current ( $D_{1}$ ), Pulse on Time ( $D_{2}$ ), Percentage of Reinforcement ( $D_{3}$ ), and Flushing Pressure ( $D_{4}$ ) have been selected to obtains the most suitable alternative. The benefit criterion was MRR ( $C_{1}$ ), and the cost criteria were EWR ( $C_{2}$ ) and Surface Roughness ( $C_{3}$ ). This decision making (shown in Figure 7) have been demonstrated using a hierarchical model as shown in Figure 8.

Figure 7.

Decision-making module.

Figure 8.

Decision making by hierarchical structure.

The computational procedure involved in TOPSIS is summarized as follows –

Step 1: The decision matrix is delineated in Table 8.

Table 8.

Structure for decision matrix.

Sl. no.	$C_{1}$	$C_{2}$	$C_{3}$	Sl. no.	$C_{1}$	$C_{2}$	$C_{3}$
1	0.0999	0.0028	7.7919	16	0.4998	0.0028	16.8724
2	0.1619	0.0041	10.5094	17	0.1057	0.001	10.9279
3	0.4156	0.0013	10.7293	18	0.6248	0.0028	13.8752
4	0.6248	0.0028	13.8752	19	0.4867	0.0108	16.7853
5	0.0751	0.0032	8.9826	20	0.1049	0.0016	10.6884
6	0.1698	0.0192	12.8739	21	0.0875	0.0009	9.1284
7	0.3856	0.0029	12.9837	22	0.1893	0.0037	11.8073
8	0.4867	0.0108	16.7853	23	0.0985	0.0061	8.829
9	0.1006	0.0047	7.8934	24	0.3928	0.0012	11.9719
10	0.1721	0.0051	11.1073	25	0.1142	0.0017	9.2943
11	0.3554	0.0015	10.9973	26	0.0929	0.0023	11.0981
12	0.6843	0.0019	14.2872	27	0.1011	0.0032	10.6172
13	0.0887	0.0034	9.0183	28	0.1092	0.0014	10.9891
14	0.1997	0.012	12.8641	29	0.1049	0.0021	10.6503
15	0.3891	0.0025	13.4281	30	0.1042	0.0033	10.5173

Step 2: The normalized decision matrix R is calculated following equation (2) and is depicted in Table 9.

Table 9.

Normalized decision matrix.

Sl. no.	$R_{i 1}$	$R_{i 2}$	$R_{i 3}$
1	0.057436751	0.089507133	0.1199873
2	0.093083183	0.131064016	0.161834024
3	0.238946083	0.041556883	0.165220259
4	0.359224044	0.089507133	0.213663906
5	0.043178178	0.102293866	0.138322864
6	0.097625228	0.613763195	0.198244908
7	0.221697809	0.092703816	0.199935716
8	0.279824491	0.345241797	0.258476473
9	0.057839211	0.150244115	0.121550296
10	0.098947596	0.163030849	0.171041073
11	0.204334547	0.04795025	0.169347185
12	0.39343312	0.060736983	0.220008285
13	0.050997395	0.108687232	0.138872607
14	0.114816008	0.383601997	0.198093998
15	0.223710108	0.079917083	0.206779022
16	0.287356238	0.089507133	0.259817724
17	0.060771417	0.031966833	0.168278496
18	0.359224044	0.089507133	0.213663906
19	0.279824491	0.345241797	0.258476473
20	0.060311463	0.051146933	0.164590441
21	0.050307465	0.02877015	0.140568035
22	0.108836606	0.118277282	0.181820358
23	0.056631832	0.194997682	0.135957581
24	0.225837395	0.0383602	0.18435503
25	0.065658428	0.054343616	0.143122725
26	0.053412154	0.073523716	0.170899403
27	0.058126682	0.102293866	0.163494034
28	0.062783716	0.044753566	0.169220914
29	0.060311463	0.067130349	0.16400374
30	0.059909003	0.105490549	0.161955676

Step 3: The decision matrix for weighted normalized data is calculated by considering the weight calculated through AHP and is shown in Table 10.

Table 10.

Weighted normalized decision matrix.

Sl. no.	$W_{i 1}$	$W_{i 2}$	$W_{i 3}$
1	0.041354	0.015216	0.013199
2	0.06702	0.022281	0.017802
3	0.172041	0.007065	0.018174
4	0.258641	0.015216	0.023503
5	0.031088	0.01739	0.015216
6	0.07029	0.10434	0.021807
7	0.159622	0.01576	0.021993
8	0.201474	0.058691	0.028432
9	0.041644	0.025541	0.013371
10	0.071242	0.027715	0.018815
11	0.147121	0.008152	0.018628
12	0.283272	0.010325	0.024201
13	0.036718	0.018477	0.015276
14	0.082668	0.065212	0.02179
15	0.161071	0.013586	0.022746
16	0.206896	0.015216	0.02858
17	0.043755	0.005434	0.018511
18	0.258641	0.015216	0.023503
19	0.201474	0.058691	0.028432
20	0.043424	0.008695	0.018105
21	0.036221	0.004891	0.015462
22	0.078362	0.020107	0.02
23	0.040775	0.03315	0.014955
24	0.162603	0.006521	0.020279
25	0.047274	0.009238	0.015743
26	0.038457	0.012499	0.018799
27	0.041851	0.01739	0.017984
28	0.045204	0.007608	0.018614
29	0.043424	0.011412	0.01804
30	0.043134	0.017933	0.017815

Step 4: The ideal solutions (positive and negative) along with the distance of every alternative to PIS and NIS is calculated. The PIS and NIS are shown in Table 11, and the space of every alternative is shown in Table 12.

Table 11.

PIS and NIS of criteria.

	$C_{1}$	$C_{2}$	$C_{3}$
PIS	0.283272	0.004891	0.013199
NIS	0.031088	0.10434	0.02858

Table 12.

Distance of each alternative.

Sl. no.	Distance from PIS	Distance from NIS
1	0.242138	0.091022
2	0.216999	0.090227
3	0.111363	0.171576
4	0.028626	0.244436
5	0.252501	0.087971
6	0.235213	0.039783
7	0.124437	0.15624
8	0.099083	0.176394
9	0.242509	0.080944
10	0.213328	0.087058
11	0.136298	0.151046
12	0.012271	0.269174
13	0.246936	0.08707
14	0.209653	0.065096
15	0.122881	0.158638
16	0.07859	0.197108
17	0.239576	0.10022
18	0.028626	0.244436
19	0.099083	0.176394
20	0.239928	0.097004
21	0.247061	0.100441
22	0.205586	0.096972
23	0.244144	0.073127
24	0.120887	0.164114
25	0.236052	0.097319
26	0.244997	0.092654
27	0.241791	0.088252
28	0.238145	0.098263
29	0.239985	0.094333
30	0.240536	0.087904

Step 5: The relative proximity of each alternative to the positive ideal solution and the negative ideal solution is evaluated and ranked accordingly. The ranked alternatives are shown in Table 13.

Table 13.

Ranking of alternative.

Sl. no.	RC_i
12	0.956399
4	0.895166
18	0.895166
16	0.714942
8	0.640322
19	0.640322
3	0.606407
24	0.575836
15	0.563507
7	0.556653
11	0.525662
22	0.320507
17	0.294942
2	0.293683
28	0.292095
25	0.291925
10	0.289819
21	0.289038
20	0.287904
29	0.282166
26	0.274406
1	0.273208
30	0.26764
27	0.267395
13	0.260683
5	0.258379
9	0.25025
14	0.236928
23	0.230486
6	0.144666

As per the TOPSIS method, the best alternative was found to be the 12th alternative because of its highest proximity to the ideal solution.

In correspondence to the VIKOR technique, the alternatives are denoted as, and the steps involved in VIKOR are as follows –

Step 1: The best and worst alternatives of all criteria functions are shown in Table 14.

Table 14.

Best and worst alternatives.

	$C_{1}$	$C_{2}$	$C_{3}$
$F_{i}^{+}$	0.6843	0.0192	16.8724
$F_{i}^{-}$	0.0751	0.0009	7.7919

Step 2: The $S_{i}$ and $R_{i}$ are calculated following equations (8) and (9) and is depicted in Table 15.

Table 15.

$S_{j}$ and $R_{j}$ .

Alternative no.	$C_{1}$	$C_{2}$	$C_{3}$	$S_{j}$	$R_{j}$
1	0.0999	0.0028	7.7919	0.70834	0.690689
2	0.1619	0.0041	10.5094	0.680059	0.617413
3	0.4156	0.0013	10.7293	0.35687	0.317571
4	0.6248	0.0028	13.8752	0.161664	0.073692
5	0.0751	0.0032	8.9826	0.75579	0.72
6	0.1698	0.0192	12.8739	0.839639	0.608076
7	0.3856	0.0029	12.9837	0.434499	0.353027
8	0.4867	0.0108	16.7853	0.434451	0.233539
9	0.1006	0.0047	7.8934	0.726392	0.689862
10	0.1721	0.0051	11.1073	0.684537	0.605358
11	0.3554	0.0015	10.9973	0.433123	0.38872
12	0.6843	0.0019	14.2872	0.087973	0.078683
13	0.0887	0.0034	9.0183	0.742007	0.703926
14	0.1997	0.012	12.8641	0.737297	0.572738
15	0.3891	0.0025	13.4281	0.43203	0.34889
16	0.4998	0.0028	16.8724	0.345707	0.218056
17	0.1057	0.001	10.9279	0.722753	0.683835
18	0.6248	0.0028	13.8752	0.161664	0.073692
19	0.4867	0.0108	16.7853	0.434451	0.233539
20	0.1049	0.0016	10.6884	0.726371	0.68478
21	0.0875	0.0009	9.1284	0.721535	0.705345
22	0.1893	0.0037	11.8073	0.659683	0.58503
23	0.0985	0.0061	8.829	0.753213	0.692344
24	0.3928	0.0012	11.9719	0.39794	0.344517
25	0.1142	0.0017	9.2943	0.69942	0.673789
26	0.0929	0.0023	11.0981	0.752019	0.698963
27	0.1011	0.0032	10.6172	0.744863	0.689271
28	0.1092	0.0014	10.9891	0.723073	0.679698
29	0.1049	0.0021	10.6503	0.730554	0.68478
30	0.1042	0.0033	10.5173	0.740918	0.685607

Step 3: The $Q_{i}$ is calculated using equation (10) and is shown in Table 16

Table 16.

Value of $Q_{i}$ .

Alternative no.	$C_{1}$	$C_{2}$	$C_{3}$	$Q_{i}$
1	0.0999	0.0028	7.7919	0.889986
2	0.1619	0.0041	10.5094	0.814485
3	0.4156	0.0013	10.7293	0.367538
4	0.6248	0.0028	13.8752	0.049019
5	0.0751	0.0032	8.9826	0.944225
6	0.1698	0.0192	12.8739	0.913413
7	0.3856	0.0029	12.9837	0.446606
8	0.4867	0.0108	16.7853	0.354135
9	0.1006	0.0047	7.8934	0.901354
10	0.1721	0.0051	11.1073	0.808138
11	0.3554	0.0015	10.9973	0.473303
12	0.6843	0.0019	14.2872	0.003861
13	0.0887	0.0034	9.0183	0.922621
14	0.1997	0.012	12.8641	0.817998
15	0.3891	0.0025	13.4281	0.441763
16	0.4998	0.0028	16.8724	0.283126
17	0.1057	0.001	10.9279	0.89427
18	0.6248	0.0028	13.8752	0.049019
19	0.4867	0.0108	16.7853	0.354135
20	0.1049	0.0016	10.6884	0.897408
21	0.0875	0.0009	9.1284	0.910101
22	0.1893	0.0037	11.8073	0.775878
23	0.0985	0.0061	8.829	0.921115
24	0.3928	0.0012	11.9719	0.415704
25	0.1142	0.0017	9.2943	0.870978
26	0.0929	0.0023	11.0981	0.925441
27	0.1011	0.0032	10.6172	0.913183
28	0.1092	0.0014	10.9891	0.891283
29	0.1049	0.0021	10.6503	0.900191
30	0.1042	0.0033	10.5173	0.907725

Step 4: All the alternatives were ranked in ascending order as per their $S_{i}$ , $R_{i}$ , and $Q_{i}$ values and is shown in Table 17. The lower the value is, the more optimal is the solution.

Table 17.

Ranking of alternatives as per VIKOR.

Alternative no.	$Q_{i}$	Alternative no.	$S_{i}$	Alternative no.	$R_{i}$
12	0.003861	12	0.087973	4	0.073692
4	0.049019	4	0.161664	18	0.073692
18	0.049019	18	0.161664	12	0.078683
16	0.283126	16	0.345707	16	0.218056
8	0.354135	3	0.35687	8	0.233539
19	0.354135	24	0.39794	19	0.233539
3	0.367538	15	0.43203	3	0.317571
24	0.415704	11	0.433123	24	0.344517
15	0.441763	8	0.434451	15	0.34889
7	0.446606	19	0.434451	7	0.353027
11	0.473303	7	0.434499	11	0.38872
22	0.775878	22	0.659683	14	0.572738
10	0.808138	2	0.680059	22	0.58503
2	0.814485	10	0.684537	10	0.605358
14	0.817998	25	0.69942	6	0.608076
25	0.870978	1	0.70834	2	0.617413
1	0.889986	21	0.721535	25	0.673789
28	0.891283	17	0.722753	28	0.679698
17	0.89427	28	0.723073	17	0.683835
20	0.897408	20	0.726371	20	0.68478
29	0.900191	9	0.726392	29	0.68478
9	0.901354	29	0.730554	30	0.685607
30	0.907725	14	0.737297	27	0.689271
21	0.910101	30	0.740918	9	0.689862
27	0.913183	13	0.742007	1	0.690689
6	0.913413	27	0.744863	23	0.692344
23	0.921115	26	0.752019	26	0.698963
13	0.922621	23	0.753213	13	0.703926
26	0.925441	5	0.75579	21	0.705345
5	0.944225	6	0.839639	5	0.72

It was found that the alternative number 12 was the best alternative as per the ranking of $S_{i}$ and $Q_{i}$ . While according to the ranking of $R_{i}$ it was found that the best alternatives were 4th, 18th, and 12th in the order. It has been observed that the 12th alternative satisfies both the conditions for a compromise solution. It has an adequate advantage and as suitable constancy in making decision by consensus. Hence, the optimal alternative is 12th alternatives. The illustrative comparison of VIKOR and TOPSIS methods are shown in Figure 2. Accordingly, the alternative having highest relative closeness coefficient was considered as the optimal solution; and as per VIKOR method, the compromise solution is the optimal solution (i.e. being nearest to the ideal solution, hence the lowest $Q_{i}$ and $S_{i} / R_{i}$ values). Therefore, it was also concluded from the graph that the Alternative 12 has the optimal solution for both as per the TOPSIS and VIKOR approach. It can also be found that alternative 4 and alternative 18 came just below the best solution with both methods.

Comparison of alternatives

A confirmatory test becomes essential to validate the TOPSIS and VIKOR results obtained. The parameters combination for Alternative 12, which has observed to be the most suitable solution as directed by both the applied methods, was considered. Similarly to ensure that the result was optimal, alternative 18 and 4 were also chosen. The outcomes are summarized in Table 18.

Table 18.

Judgment of alternatives.

Alternative	Experimental result
	MRR	EWR	SR
A12	0.6843	0.0019	14.2872
A4	0.6248	0.0028	13.8752
A18	0.6248	0.0028	13.8752
Improvement of A12
°With respect to A4	0.0595	−0.0009	0.412
°With respect to A18	0.0595	−0.0009	0.412
°With respect to A4 (in %)	9	−47	3
°With respect to A18 (in %)	9	−47	3

In machining of AMCs it is essential to have good MRR. It was observed that the surface roughness of AMCs would have been compromised against of rest performance measures because of the presence of non-conductive cenospheres. Using optimum machining condition (selected) there is an improvement of 9% in MRR value and 47% less electrode wear rate has been obtained. The above is considerable improvement in the experimental results which shows the effectiveness of TOPSIS and VIKOR process.

Conclusions

Successful implementation and thereby prediction of optimum processing by multi criteria decision-making approach have found to be less efficient due to lack of real time data and other machining discrepancies mean while prejudice. It has been identified that multiple criteria issues during machining adhere to imprecise and uncertain data and the methods applied. The present approach has been proposed followed by AHP technique to calculate the criteria weight and applied it to both the TOPSIS and VIKOR methods for the newly prepared AA6061-cenosphere AMCs. The decision-making module was constructed with a committee of four decision makers, and three different criteria is evaluated the results. Hence all the alternatives were ranked accordingly by both the methods. It can be concluded that both the applied MCDM approaches, TOPSIS and VIKOR accrue the similar possible optimal solution (alternative 12) with process parameters of pulse current 10 A, pulse on time 1010 µs, percentage of reinforcement as 2% and flushing pressure as 0.6 MPa. Hence, in EDM processing conditions, both TOPSIS and VIKOR accumulate the same result as desired. It has been concluded that the computational effort applied to obtain the optimal processing conditions for ED machining of the AMCs are efficient and effective. The generated results and approach have been discussed would helps by providing the necessary database and guidelines for obtaining optimal processing conditions in the manufacturing industry. As the multi-criteria problems adhere to uncertain and imprecise data, therefore the current study has a further scope of optimization through fuzzy extended MCDM techniques.

Highlights

Compo-casting processing route was used to fabricate the cenosphere reinforced AMC.

EDM has been employed to investigate the machinability of the prepared AMC.

Influence of process variables can be estimated by TOPSIS and VIKOR based MCDM approach.

Novelty statement

Although there were several study has been reported relating to optimization of process parameters by TOPSIS and VIKOR implementing along/or in conjunction with other techniques so as to further furnishing of the optimized results. This study in particular for obtaining the optimum processing condition while examine the machinability of newly prepared AMC’s. The nonconductive cenosphere particles may barricades the machining process by declining the electrical conductivity of AMC with increasing its percentage incorporation. The experiments explore the difficulties obtaining during machining and thereby select an optimal processing condition.

Novelty of the study:

This study in particular to obtain the optimal procession condition implementing TOPSIS in conjunction with VIKOR technique during machining through EDM.

The incorporation of cenosphere particles leads to lower electrical conductivity of the AMC’s during machining with EDM which alter the machining operation has making the observation in particular.

Adverse characteristics of machining for the newly prepared AMC’s along with selection of optimal processing condition for conflicting multiple responses incorporating combination of different MCDM approaches is the prime novelty of the research.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Abhijit Dey

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Optimum performance evaluation during machining of Al6061/cenosphere AMCs using TOPSIS and VIKOR based multi-criteria approach

Abstract

Keywords

Introduction

Materials and methods

Experimental details

Performing experiments

Parametric analysis using TOPSIS and VIKOR

Weight calculation through AHP

Results and discussions

Comparison of alternatives

Conclusions

Highlights

Novelty statement

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References