Application of Type-2 Fuzzy AHP-ARAS for Selecting Optimal WEDM Parameters

: Machining of the nickel-based alloy is very demanding due to its extreme mechanical prop-erties, for example, higher fatigue strength, better corrosion and creep resistance feature, substantial work hardening capability, and appreciable tensile and shear strength. Owing to these properties, the selection of machining parameters is a major challenge for modern machining industries. Therefore, the present experimental work is carried out to select the best parametric combination of the wire electrical discharge machining (WEDM) machine for reducing machining cost and human effort. The Trapezoidal Interval Type-2 fuzzy number (T2FS) integrated Analytical Hierarchy Process (AHP)-based Additive Ratio Assessment (ARAS) method is used for selecting the best WEDM process parameters of Inconel-800 superalloy. Finally, the results were compared with some existing multicriteria decision-making methods to conﬁrm the validity of the adopted method. The comparison shows that Type-2 Fuzzy AHP-ARAS synergy can help to formulate the problem and facilitate the assessment and ranking of WEDM process parameters when multiple criteria are jointly considered.


Introduction
Owing to their extreme mechanical properties, for instance high strength and creepcorrosion resistance, chromium-nickel-based alloys possess very dull machinability [1,2]. However, these alloys are useful in the aerospace, aviation and nuclear industries, where the elevated temperature is the primary concern [3][4][5]. Moreover, electric discharge machining (EDM) is a process of repetitive sparking cycles. A series of electrical pulses generated by the pulse generator unit is applied between the workpiece and the traveling wire electrode. In the event of spark discharge, there is a flow of current across the wire electrode-workpiece gap. The energy content of a single spark discharge can be expressed as a product of pulse on time with the peak current. The energy contained in a tiny spark discharge removes a fraction of workpiece material. A large number of such timespaced tiny discharges between the workpiece and wire electrode causes the electroerosion of the workpiece material. Primarily, electric discharge machining exists in the form of die-sinking machines and in the 1960s, the wire electrode cut type of machines were developed for the purpose of making tools (dies) from hardened steel. In fact, wire electrical discharge machining is a technological advancement in non-traditional machining processes, where traveling wear removes the materials from the workpiece. This non-conventional machining is very useful to cut electrically conductive materials by compared with some entrenched uncertainty based MCDM methods like fuzzy TOPSIS and Fuzzy VIKOR. Chatterjee and Bose (2013) [40] developed a hybrid fuzzy-based ARAS method for ranking the vendors for a wind farm, which was used the following year by Barak et al. (2014) [25] for the selection of hydraulic fracturing treatment. However, from the above literature review, it was manifested that very few manuscripts were published on the hybrid type-2 Fuzzy-ARAS approach in the manufacturing domain. Therefore, the objective of the current study is to develop a Fuzzy coupled MCDM approach for the problem in question. The results of this developed method are equated with some prevailing methods to report the validity of T2F-AHP-ARAS.
The selection of the optimal values of different process parameters of WEDM is very important for enhancing machining performances. Several mathematical approaches, such as gray relational analysis, Pareto optimality, desirability function, simulated annealing, etc., have already been successfully implemented by researchers. However, most of the time, the researchers have found sub-optimal solutions. Thus, an ideal backdrop was created to explore the efficacy of an integrated Fuzzy coupled MCDM model to minimize the ambiguity and uncertainty of the criteria weights. The quality of the developed model will be enhanced when type-2 Fuzzy logic is coupled with a prominent ARAS model. Whereas type-2 Fuzzy logic handled the uncertainties in the values of the membership function, ARAS obtained the best possible solution. Thus, in the present study, a mathematical endeavor was undertaken to select the optimal parametric combination of the WEDM machine.

Materials and Measuring Equipment
The cutting operation was carried out on a WEDM machine (ELPULS-40 A DLX). Taguchi's L18 orthogonal array was used as the design of the experiments. A 0.25-mm thick brass wire was used for cutting. De-ionized water was used as a di-electric fluid. Commercially available Inconel 800 was used as a workpiece. The variable WEDM parameters were selected after an extensive literature review and are shown in Table 1; the values of fixed machining parameters are shown in Table 2. Furthermore, the surface roughness of the cutting zone was measured by a 3D profilometer. Cutting time was measured by a digital stopwatch. The MRR (gm/min) is calculated by utilizing Equation (1). The width of kerf (mm) was observed with an optical microscope. To find out the power consumption (kWh) during the cutting, Equation (2) is used. Finally, for observing the load power, a 3 Ø wattmeter was coupled to a transformer. Figure 1 shows the spark event on WEDM during the metal cutting operation.

Preliminaries
This section gives a brief description of the key concepts related to the T2FS and the ARAS method is given.

Fuzzy Sets
In the universe of discourse , is termed as fuzzy sets , if it is typified by membership value that maps every element of to a real-valued number in 0, 1 .
where denotes the membership value of ∈ .

Type-2 Fuzzy Sets
In the simplest language, the T2FS can be defined as the blurriness of T1F membership functions [41]. If is a T2FS on the universal discourse of then it is defined as follows: = 〈 , ; , 〉 ∀ ∈ , ∀ ∈ ⊆ 0, 1 where , represent the secondary membership grade such that 0 < , < 1.

Interval Type-2 Fuzzy Sets
IT2FS is a special case of T2FS. For the condition ∀ , = 1, then the T2FS is called IT2FS [42]. The IT2FS can be described as follows:

Preliminaries
This section gives a brief description of the key concepts related to the T2FS and the ARAS method is given.

Fuzzy Sets
In the universe of discourse U, A is termed as fuzzy sets (FS), if it is typified by membership value (µ A ) that maps every element of U to a real-valued number in [0, 1].
where µ A (x) denotes the membership value of x ∈ U.

Type-2 Fuzzy Sets
In the simplest language, the T2FS can be defined as the blurriness of T1F membership functions [41]. If A is a T2FS on the universal discourse of X then it is defined as follows: where µ A (x, u) represent the secondary membership grade such that 0 < µ A (x, u) < 1. A can be conveyed as

Interval Type-2 Fuzzy Sets
IT2FS is a special case of T2FS. For the condition ∀µ A (x, u) = 1, then the T2FS A is called IT2FS [42]. The IT2FS can be described as follows:

Footprint of Uncertainty
The footprint of uncertainty (FOU) [43] can be defined as the two-dimensional support of the secondary membership grade. It is the union of all primary membership grades. FOU is often described by upper membership function ( A U ) and lower membership function ( A L ) where A U and A L are T1FS.

Trapezoidal Interval Type-2 Fuzzy Number
An interval type-2 fuzzy number is termed a trapezoidal interval type-2 fuzzy number (TrI2N) (Figure 2), if A U and A L are type-1 trapezoidal fuzzy number [21]. If A is a TrI2N, then it can be defined as follows:

Footprint of Uncertainty
The footprint of uncertainty (FOU) [43] can be defined as the two-dimensional support of the secondary membership grade. It is the union of all primary membership grades. FOU is often described by upper membership function and lower membership function where and are T1FS.

Ranking of TrI2N
Previous authors have developed a ranking algorithm for T2FS, which is utilized to develop an extension of the fuzzy TOPSIS method [44]. Further, in [17], the extended fuzzy TOPSIS method was applied in a decision-making approach whereby the alternatives are assessed in the form of linguistic terms on the basis of the criteria. The ranking of a TrI2N is the difference of the basic ranking score and the average of the standard deviation of the A. The basic ranking score of A is denoted by Br( A) which is calculated as follows: The average of the standard deviation of A is denoted by sd( A) which is calculated as follows: where The rank of A is calculated as follows:

Proposed TrI2N Analytical Hierarchy Process (AHP) Integrated ARAS Method
Selecting the best WEDM parameters settings on the basis of the performance measures is a case of multi-criteria decision making (MCDM). The performance parameters based on which the decision is taken are called criteria. The degree to which a criterion influences the section of the best WEDM parameters is termed the weightage of the criteria [45]. Computation of the criteria weights is done by applying the fuzzy integrated analytical hierarchy process (AHP). The proposed ranking method is the integration of the AHP and ARAS method. Conferring to the ARAS method, the profitability function, which is helpful in selecting the decision alternatives, is proportional to the relative effect of values and criteria weights. The weightage of the criteria is the degree to which the criteria affects the final decision. The steps for the proposed algorithm is as follows: Step 1. Formation of the decision matrix The value of the performance measures as obtained from the experimental design forms the decision matrix. The number of designed experiments is the alternatives, and the number of performance measures based on which the decision is to be taken are the criteria. If there are 'm' alternatives and 'n' criteria, then the decision matrix (D) is represented as Step 2: Addition of the idle best value in the decision matrix When the best solution is unknown, for a benefit criterion, the maximum value is always preferred and vice-versa for a non-benefit criterion. The main idea is to create a virtual best alternative with respect to which all other feasible alternatives are compared. The matrix is called the initial decision matrix.
Step 4: Computation of weighted normalized decision matrix Step 5: Computation of weights of the criteria AHP is an effective as well as efficient decision support system tool that helps to recognize and define a problem in detail. It breaks down the problem into its constituent parts, which are then structured hierarchically. One of the advantages of using AHP is that it allows the decision makers (DM) to subjectively assess the alternatives on the basis of the criteria. Moreover, due to the existence of uncertainty in human psychology, DMs favors assessing the alternatives subjectively. Meanwhile, it is worth pointing out that in order to reach a precise decision, it is crucial to consider the views of more than one decision maker (DM). Decision-making problems which involve more than one DM are called multi-criteria group decision making (MCGDM). The 1-9 scale defined by Saaty to quantify the subjective assessment [46] fails to aggregate the views in a scenario in which decision is taken by a group of DMs [47,48]. With the development of fuzzy sets (FSs), it is applied to quantify the subjective assessment of the criteria. The major advantage of using fuzzy logic for quantifying the subjective assessment is its ability to aggregate the views of different DMs in a group decision-making environment [41]. However, on the basis of the falsificationism concept of Karl Popper [49], Mendel argued that the application of interval type-2 fuzzy sets for quantifying the subjective assessment is more scientifically correct than the application of general fuzzy logic [50,51]. Hence, in this paper, the subjective assessment of the criteria is quantified using interval type-2 fuzzy sets which are then integrated with AHP for computing the weightage of the criteria. The steps applied for computing the weights of any criteria are described below.
Step 5.1: Formation of aggregated pairwise comparison matrix: For computing the weights, the views of the three decision-makers are integrated. The aggregation of the pairwise comparison matrices is done to incorporate the knowledge of decision makers of different backgrounds. The aggregation is done according to the interval value aggregating operator as discussed in the literature [41], which is done as follows: where a k ij indicates the kth decision maker's preference of i th criterion over j th criterion. Considering that the aggregated pair-wise comparison matrix is represented by A k , then The linguistic ratings used by the decision makers for assessing the criteria and their corresponding TrI2N are revealed in Table 3. The aggregated pair-wise comparison matrix formulated according to Equation (20) is demonstrated in Table 4.  Step 5.2: Calculation of column-wise geometric mean (G m ) The geometric mean of fuzzy comparison values is calculated as , (j = 1, 2, 3, · · · , n) The column-wise geometric mean of the aggregated pairwise comparison matrix is shown in Table 5. Step 5.3: Calculation of fuzzified weights ( w j f ) The fuzzified weight is the normalized fuzzy values of the geometric mean of fuzzy comparison values of each criterion.
The fuzzified weights as computed by the Equation (22) is shown in Table 6. Step 5.4: Computation of the weights of the criteria The weights of the criteria are computed according to the Equation (23).
The computed weights of the criteria are shown in Table 7. Step 6: Calculation of the optimality function Optimality function (S i ) is the sum of all the weighted normalized values of an alternative for the different criteria, which is computed according to Equation (24).
Step 7: Ordering and Ranking of Alternatives The alternative with the maximum value of optimality function is the most effective and is ranked the first, and the other alternatives are ranked on the basis of descending value.

Results and Discussions
In this section, the result acquired after employing the proposed TrI2N integrated AHP-ARAS method is discussed.

Computation of the Best Cutting Parameters
A total of 18 experiments were designed, as shown in Table 8, and the result obtained after conducting the experiments, in the form of a decision matrix, are shown in Table 9.
After the formation of the decision matrix, in the next step, the best solution for each criterion is computed and added as the zeroth experiment in the decision matrix to form the initial decision matrix, which is shown in Table 10.  After that, the normalized decision matrix is calculated according to Equation (18). The computed weights of the machining performances are shown in Table 7. Then, the weights are multiplied with the normalized decision elements to form the weighted normalize decision matrix, which is presented in Table 11.
In the next step, the value of the optimality function for every alternative is calculated, conferring to Equation (24). Then, the weightage of the optimality function for each alternative with regard to the zeroth alternative is computed according to Equation (25). The reason for the comparison between the optimality function for each alternative with the zeroth alternative is that it is considered to be the idle solution. Lastly, alternatives are ranked in descending order of the weightage of the alternatives, except for the zeroth experiment. Table 12 shows the ranking of the alternatives.

Comparison of Results
This subsection implements a comparative assessment with other standards to validate the effectiveness and applicability of the projected methods and observe the efficiency of the proposed scale. Although this paper presents the first integrated TrI2N based AHP-ARAS method, there are many entrenched TrI2N-based MCDM approaches. To verify the effectiveness and efficiency of the projected technique as well as scale, the result is compared with the results obtained from [18][19][20][21][22].

Conclusions
Advancement in the nuclear, aerospace, oil and gas, automotive and marine industries has created a need for material with high material strength and less corrosion. The quest for such a material ended with nickel-based alloys named Inconel-800. Although Inconel-800 has excellent mechanical and chemical properties, its machinability is poor. Hence, the non-traditional machining method is the best and most economical way for machining the Inconel-800 superalloy. Out of all the NTM processes, WEDM is the most widely used machining technique for difficult-to-machine materials because of its capacity to produce jobs with minute accuracy and precision. The contributions and findings drawn from the analysis are as follows: • To improve the machining endeavor and to reduce machining expenses, optimum machining parameters selection is a crucial concern in the manufacturing domain.

•
In this paper, an interval type-2 fuzzy-integrated AHP-ARAS method is proposed. In the method, the best WEDM parameter settings are selected by applying the ARAS ranking method and the weightage of the criteria are computed using the AHP method.

Conclusions
Advancement in the nuclear, aerospace, oil and gas, automotive and marine industries has created a need for material with high material strength and less corrosion. The quest for such a material ended with nickel-based alloys named Inconel-800. Although Inconel-800 has excellent mechanical and chemical properties, its machinability is poor. Hence, the non-traditional machining method is the best and most economical way for machining the Inconel-800 superalloy. Out of all the NTM processes, WEDM is the most widely used machining technique for difficult-to-machine materials because of its capacity to produce jobs with minute accuracy and precision. The contributions and findings drawn from the analysis are as follows:

•
To improve the machining endeavor and to reduce machining expenses, optimum machining parameters selection is a crucial concern in the manufacturing domain.

•
In this paper, an interval type-2 fuzzy-integrated AHP-ARAS method is proposed. In the method, the best WEDM parameter settings are selected by applying the ARAS ranking method and the weightage of the criteria are computed using the AHP method.

•
Based on the concept of falsificationism, application of interval type-2 fuzzy sets for quantifying the subjective assessment is more scientifically correct than the application of general fuzzy logic. Hence, interval type-2 fuzzy numbers are applied for handling the uncertainties associated with the subjective assessment of the criteria. • The proposed model computed the best WEDM parameter settings for machining Inconel-800 superalloy is pulse-on time = 105 µs, pulse-off time = 57 µs, peak current = 210 A and spark gap voltage = 50 v. • For validation purposes, the results of the adopted method were extensively compared with some existing methods proposed by the previous researchers from literature [18][19][20][21][22]. The comparison shows that the results of the TrI2N AHP-ARAS approach are reasonably consistent with the other approaches, which shows the applicability of the proposed approach.