Solution ranking

Since MOPSO results in a large number of non-dominated solutions, choosing a best solution depends on decision maker‟s judgement and intuition. Usually, multi-attribute decision making (MADM) approaches are adopted to obtain scores for the solutions and the solution exhibiting maximum score is selected as the best one. However, the weights

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assigned in multi-attribute decision making process for converting multiple objectives into a single equivalent objective score are reasonably subjective in nature and affect the decision of ranking the alternative solutions considerably. In order to avoid uncertainty of subjective assignment of weights by the experts and extract the accurate information from the available data, maximum deviation theory (MDT) suggested by Wang (1998) is adopted in this work. The basic idea of MDT rests on smaller weight should be assigned to the attribute having similar values in comparison to the attribute having larger deviations.

The non-dominated solutions obtained in MOPSO solutions are used as the decision matrix. Every element of the decision matrix denotes the value of jth attribute for ith alternative where i=1, 2 ...n, and j=1, 2...m. Normalization of each attribute is carried out to transform different scales and units among various attributes into a common measurable scale. The normalization of the attribute depends on its type such as “higher the better” and “lower the better”. The following equations are used for normalization of attributes.

x_ij∗ ₌ maxi xij −xij

maxi xij −mini xij , for lower the better attributes (3.11)

x_ij∗ = xij−min xij

maxi xij −mini xij , for higher the better attributes (3.12)

The difference of performance values for each alternative is computed. For the attribute {Aj j=1, 2…m}, the deviation value of the alternative {Si| i = 1, 2 ….n} from all the other

alternatives can be computed by the following equation

Dij wj = Ni=1d r ij, r lj wj _(3.13)

wherewj is the weight of the attributes to be calculated and Dij(wj) is the deviation value of

the alternatives.

The total deviation values of all alternatives with respect to other alternatives for the attribute {Aj| j =1, 2… m} can be computed by the following relation.

Dj wj = Ni=1Dij wj = i=1N Nl=1d r ij, r lj wj

(3.14) whereDj(wj) is the total deviation value of all the alternatives.

The deviation of all the attributes along all the alternatives can be calculated by the relation D wj = j=1M Dj wj = Mj=1 Ni=1 Nl=1d r ij, r lj wj

(3.15) where D(wj) is deviation of all the attributes along all the alternatives.

A linear programming model is constructed for finding out the weight vector w to maximize all deviation values for all the attributes and is given by

34 D wj = Mj=1 Ni=1 Nl=1d r ij, r lj wj

s. t Mj=1w_j2= 1, wj ≥ 0, j = 1,2, … … , M (3.16) A Lagrange function is constructed for solving the above model

L wj, α = j=1M Ni=1 Nl=1d r ij, r lj wj+ α Mj=1wj2− 1

(3.17) whereis the Lagrange multiplier. The partial derivative of L (wj,) with respect to wj and 

are ∂L dwj= d r ij, r lj N l=1 N i=1 + 2 ∝ wj = 0 ∂L d∝= wj 2_{− 1 = 0} M j=1 (3.18) Further, wj and  values are calculated from equation 3.17 and 3.18

2 ∝= − d r Ni=1 Nl=1 ij, r lj 2 M j=1 w_j = Ni=1 Nl=1d r ij,r lj Ni=1 Nl=1d r ij,r lj 2 M j=1 (3.19)

The normalized attribute weights can be further determined by the following relation wj = d r ij,r lj N l=1 N i=1 d r ij,r lj N l=1 N i=1 M j=1 (3.20) The non-dominated solutions obtained through MOPSO algorithm are ranked by estimating the composite score of each solution by addition of the weighted performance of all attributes. Considering the ranking of the solutions, the tool engineer may choose suitable parametric setting from the top ranking solutions to justify the objectives set by the industry.

3.4 Materials

Inconel alloy 718, a nickel-chromium super alloy characterized by high-strength, high corrosion-resistant, good tensile and high creep rupture strength has been used as the work material in this study.A superalloy is an alloy which possesses several important features such as good mechanical strength, resistance to thermal creep deformation, good surface stability and resistance to corrosion.The super alloy is an aerospace material and has abundant usage in manufacturing of components for liquid fuled rockets, rings and casings, sheet metal parts for aircraft, land-based gas turbine engines, cryogenic tank fasteners and instrumentation parts. However, the material is extremely difficult to machine because of its low thermal conductivity, high hardness, high toughness, presence of highly abrasive

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carbide particles and strong tendency to weld to the tool to form build up edge (Kuppan et al. 2008).

The work material was supplied by Manohar Metals Private Limited, Mumbai. The chemical composition of the material is been given in Table 3.1. Table 3.2 shows the thermal property of the work material. The X-ray diffraction plot of the Inconel 718 sample used in the present study is shown in Figure 3.2. It clearly shows that there are no peaks other than γ-phase (austenite) phase which corresponds to face-centred cubic Ni-based γ- phase of Inconel 718 alloy. The sharp peak of the diffraction patterns reveals the crystalline nature of the alloy. No other peaks are observed from the XRD pattern confirming highly pure nature of the alloy.

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Table 3.1 Chemical composition of Inconel 718 sample used in the study

Chemical C% Si% Mn% S% P% Cr% Fe% Mo% Co% Nb% Cu% V% Al% Ti% W% Ni% Amount 0.039 0.027 0.032 0.005 0.008 17.21 20.143 3.121 0.086 4.989 0.009 0.015 0.568 0.816 0.214 52.739

Table 3.2 Thermal property of the Inconel 718

Properties Density Melting Temperature Thermal Conductivity Thermal expansion Possions ratio Young‟s Modulus Value 8190kg/m3 16090 K 15 W/m.K 13.0 µm/m°C 0.27-0.3 205 G Pa

37 20 40 60 80 0 1000 2000 3000 4000 5000 6000 7000 8000 220  200  In te n s it y ( a . u )

2 theta (in degree)

111



Figure 3.2 X-ray diffraction plot of the Inconel 718 work material

In EDM process the tool has to deal with a series of spark discharges. Hence, the tool must be of a good conductive material with high melting temperature, ability to withstand high temperature and dissipate the heat. Therefore, commercially available brass, copper, and graphite are used as the electrode material. The machining face of the three electrodes is cylindrical shape of diameter 13.5mm.

Experiments have been carried out in a die sinking CNC EDM machine (ECOWIN MIC- 432C) with servo-head (constant gap) as shown in Figure 3.3. The specification of the machine is given in Table 3.3.Positive polarity for electrode and side flushing is used to conduct the experiments. The EDM process is performed on Inconel 718 plate having thickness 8mm and 10×11.5 cm2 cross sectional area. A stopwatch is used to record each experimental run for 30 min. For weighing purpose, the work piece and the electrodes have been detached from the machine after each observation, cleaned and dried out. A precision electronic balance (accuracy 0.01g) is used for measuring the weights of the work piece and tool materials before and after machining. Surface roughness tester (Surftest SJ 210, Mitutoyo) is used for measuring the surface quality. A tool maker‟s microscope (Carl Zeiss) is used for measuring the crater diameter on the machined surface on the work material.

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Table 3.3 Specification of the CNC Die sinker EDM machineECOWIN MIC-432C

Mechanism of process Controlled erosion (melting and evaporation) through a series of electric spark

Spark gap 0.010- 0.500 mm

Spark frequency 200 – 500 kHz

Working Current 1-60A

Working voltage across the gap 30- 250 V Maximum Flushing Pressure 0.5 Pa Metal removal rate (max.) 680 mm3/min Specific power consumption 2-10 W/mm3/min

Dielectric fluid Liquid paraffin.

Dielectric tank Capacity

Travel limit X-axis 400mm Y-axis 400mm Z-axis 400mm