Fuzzy-MOORA Based Optimization of Machining Parameters for Machinability Enhancement of Titanium

2.1 Optimization

2.1.1 MOORA

The multi-criteria decision making based MOORA method is suitable to identify the combination of best parameters. This method was developed by Braurers and Zavadkas in the year 2004. It is used to optimize two or more conflicting objectives (criteria) subject to certain constraints [3, 6]. The reference point and ratio system are the two important elements in this method that determine the overall performance of each alternative.Ithas wide application in various sectors such as industrial sectors, manufacturing plants, banking, and insurance sectors, etc. In these aforementioned areas, multi objectives problems mostly occur where two or more conflicting attributes take place and need to identify one optimal choice [14, 15].

The proposed approach in MOORA is outlined in the following steps [3, 13]:

Step 1: MOORA method initiates with the decision matrix as shown in Equation 1 that illustrates the performance of all responses with respect to the selected input process parameters.

$P\,\,=\,\,\,\left[ \begin{matrix}   {{p}_{11}} & {{p}_{12}} & ... & ... & {{p}_{1a}}  \\   {{p}_{21}} & {{p}_{22}} & ... & ... & {{p}_{2b}}  \\   ... & ... & ... & ... & ...  \\   ... & ... & ... & ... & ...  \\   {{p}_{a1}} & {{p}_{a2}} & ... & ... & {{p}_{ab}}  \\\end{matrix} \right]$     (1)

where, P is a performance measure of the ith alternative on jth criterion and p_ij represents the output responses of the ith alternative on jth criterion, a and b are the number of alternatives and several criteria.

Step 2: The data of decision matrix gets normalized by the formation of the ration system as shown in Eq. (2).

$p_{i j}^{*}=\frac{p_{i j}}{\sqrt{\sum_{i=1}^{b} p_{i j}^{2}}}(j=1,2, \ldots \ldots, n)$     (2)

here, $p_{i j}^{*}$ represents the normalized value that lies between 0 and 1is dimensionless quantity of the i th alternative on j th criterion.

Step 3: In this step, the ranking scores are identified by MOORA index, or overall assessment values (q_i) are obtained by the addition and subtraction of weighted normalized values corresponding to each alternative shown in Eq. (3). For multi-objective optimization to measure the overall assessment values benefit response (higher-is-better) are added in normalized values in case of maximization whereas in case of minimization the non-beneficial (lower-is-better) are subtracted.

 $q_{i}=\sum_{j=1}^{x} p_{i j}^{*}-\sum_{j=x+1}^{y} p_{i j}^{*}$     (3)

where, x represents the number of criteria to be maximized belongs to benefit responses, whereas number of criteria is denoted by (y-x) which needs to be minimized. The normalized assessed value of i th alternative with respect to all criteria is represented by q_i.

Primarily, it was observed that a few of the criteria are more essential than others. Hence, in such circumstances, the more importance is given to weight criteria and it can be multiplied with the corresponding weight. In such condition Eq. (3) would be written as Eq. (4):

$q_{i}=\sum_{j=1}^{x} r_{j} p_{i j}^{*}-\sum_{j=x+1}^{y} r_{j} p_{i j}^{*}$      (4)

where, r_j is the weight of jth criteria.

Step 4: In the decision matrix, the calculated overall assessment values can be obtained positive or negative depending upon beneficial(maxima) and non-beneficial (minima) attributes. The optimal value is determined by larger MOORA value (q_i) which shows the best result and the lowest value of q_i represents the worst result.

2.1.2 Fuzzy set theory

In the real-time manufacturing system, multi-criteria decision making (MCDM) related problems occur several times, due to the presence of multiple conflicting criteria. At a large scale, these problems are more complicated because of uncertain situations. In such circumstances, Fuzzy set theory helps to treat uncertainties in the form of vagueness and ambiguity to provide the best results. In the fuzzy set theory, the linguistic approach has constructed by fuzzy logic in which variables can assume linguistic values. With the help of fuzzy set theory, the opinions given by decision makers are term as specified linguistic variables. A fuzzy membership function converts aforesaid linguistic variables into a different fuzzy number. In this way, the fuzzy set theory has the ability to solve the MCDM problems effectively with ease. Fuzzy membership function can be represented in the triangular form as shown in Figure 1. Some important definitions of fuzzy numbers and fuzzy set theory are explained below [3, 16-18]:

Definition 1: A fuzzy set $\tilde{P}$ in a universe of discourse Xis described by a membership function $\mu_{\tilde{A}}(g)$ which is characterized as the grade of membership of g in $\tilde{P}$.

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Figure 1. A triangular fuzzy membership function

Definition 2: $\tilde{P}$= (p₁, p₂, p₃), are triangular fuzzy numbers (TFNs) where $\tilde{P}$ is the membership function of the fuzzy number can be written as below (Eq. (6)):

$\mu_{\tilde{A}}(g)=\left\{\begin{array}{ll}0 & g<1, \\ \frac{g-p_{1}}{p_{2}-p_{1}} & p_{1} \leq g \leq p_{2}, \\ \frac{p_{3}-g}{p_{3}-p_{2}} & p_{2} \leq g \leq a_{3}, \\ 0 & g>p_{3}\end{array}\right.$    (5)

Definition 3: The fuzzy sum and fuzzy subtraction of two different TFNs are also triangular fuzzy numbers. But, the multiplication of two different TFNs is only an approximate TFN. For example, if there are two triangular fuzzy numbers $\tilde{P}=\left(p_{1}, p_{2}, p_{3}\right)$ and $\tilde{Q}=\left(q_{1}, q_{2}, q_{3}\right)$, and a positive real number w = (w, w, w), some important operations of fuzzy numbers can be written as follows:

$\tilde{P}(+) \tilde{Q}=\left(p_{1}+p_{2}, q_{1}+p_{2}, q_{1}+r_{2}\right)$    (6)

$\tilde{P}(-) \tilde{Q}=\left(p_{1}-p_{2}, q_{1}-q_{2}, r_{1}-r_{2}\right)$    (7)

$\tilde{P}(\times) \tilde{Q}=\left(p_{1} p_{2}, q_{1} q_{2}, r_{1} r_{2}\right)$    (8)

$\tilde{P}(/) \tilde{Q}=\left(p_{1} / q_{1}, p_{2} / q_{2}, p_{3} / q_{3}\right)$    (9)

$\tilde{P}(\times) w=\left(p_{1} w, p_{2} w, p_{3} w\right)$    (10)

Definition 4: A triangular fuzzy number $\tilde{P}=\left(p_{1}, p_{2}, p_{3}\right)$, then the defuzzified value $a(\tilde{P})$ can be determined using Eq. (11):

$a(\tilde{P})=\frac{p_{1}+p_{2}+p_{3}}{3}$     (11)

Definition 5: The two triangular fuzzy numbers are $\tilde{P}=\left(p_{1}, p_{2}, p_{3}\right)$ and $\tilde{Q}=\left(q_{1}, q_{2}, q_{3}\right)$, and the distance between $(\tilde{P})$ and $(\tilde{Q})$ can be computed using Eq. (12):

$d(\tilde{P}, \tilde{Q})=\sqrt{\frac{1}{3}\left(p_{1}-q_{1}\right)^{2}+\left(p_{2}-q_{2}\right)^{2}+\left(p_{3}-q_{3}\right)^{2}}$     (12)

Definition 6: By applying center of area approach, the best non-fuzzy performance (BNP) value can be determined and expressed as Eq. (13):

$B N P_{i}=\frac{[(r-p)+(q-p)]}{3}+p, \forall_{i}$     (13)

2.1.3 Fuzzy embedded MOORA method

The extension of the MOORA method is a systematic approach to solve the multi-criteria decision making in the fuzzy environment. Various researchers of real-time manufacturing attempted the hybridization of two approaches. Hence, this hybrid approach used to identify optimal parametric combinations to confirm improvement in the machining performance of WC cutting tool inserts. In this hybrid fuzzy-MOORA method, the opinions of decision-makers express in the terms of a set of linguistic variables. The fuzzy embedded MOORA method followed by the following steps:

Step 1: Between all alternatives(rows) and criteria (columns) the fuzzy decision matrix has been formed that belong to fuzzy triangular numbers as shown in Eq. (14)

$\tilde{P}=\left[\begin{array}{ccc}{\left[p_{11}^{a}, p_{11}^{b}, p_{11}^{c}\right]} & {\left[p_{12}^{a}, p_{12}^{b}, p_{12}^{c}\right]} & {\left[p_{1 c}^{a}, p_{1 c}^{b}, p_{1 c}^{c}\right]} \\ \ldots & \cdots & \ldots \\ {\left[p_{b 1}^{a}, p_{b 1}^{b}, p_{b 1}^{c}\right]} & p_{b 2}^{a}, p_{b 2}^{b}, p_{b 2}^{c} & p_{b c}^{a}, p_{b c}^{b}, p_{b c}^{c}\end{array}\right]$     (14)

Step 2: Using Eqns. (15-17), the normalized fuzzy decision matrix has been calculated

$p_{i j}^{a^{*}}=\frac{p_{i j}^{a}}{\sqrt{\sum_{i=1}^{b}\left[\left(p_{i j}^{a}\right)^{2}+\left(p_{i j}^{b}\right)^{2}+\left(p_{i j}^{c}\right)^{2}\right]}}$    (15)

$p_{i j}^{b^{*}}=\frac{p_{i j}^{b}}{\sqrt{\sum_{i=1}^{m}\left[\left(p_{i j}^{a}\right)^{2}+\left(p_{i j}^{b}\right)^{2}+\left(p_{i j}^{c}\right)^{2}\right]}}$     (16)

$p_{i j}^{c^{*}}=\frac{p_{i j}^{c}}{\sqrt{\sum_{i=1}^{b}\left[\left(p_{i j}^{a}\right)^{2}+\left(p_{i j}^{b}\right)^{2}+\left(p_{i j}^{c}\right)^{2}\right]}}$    (17)

Step 3: Calculate the weighted normalized fuzzy decision matrix using Eqns. (18-20).

$W_{i j}^{a}=r_{j} p_{i j}^{a^{*}}$     (18)

$W_{i j}^{b}=r_{j} p_{i j}^{b *}$     (19)

$W_{i j}^{c}=r_{j} p_{i j}^{c *}$    (20)

r_j is the weight criteria of each attribute in aforesaid equations.

Step 4: The non-fuzzy value(crisp) has converted from overall fuzzy assessment value $\left(\tilde{q}_{i}\right)$. Eq. (21) can be used to calculate the best non-fuzzy performance (BNP) as expressed below:

$B N P_{i}\left(q_{i}\right)=\frac{\left(q_{i}^{c}-q_{i}^{a}\right)+\left(q_{i}^{b}-q_{i}^{a}\right)}{3}+q_{i}^{a}$    (21)

where, $\tilde{q}_{i}=\left(q_{i}^{a}, q_{i}^{b}, q_{i}^{c}\right)$.

Step 5: In this step the overall fuzzy assessment value can be computed by applying Eq. (22).

$\tilde{q}_{i}=\tilde{W}_{i j}^{+}-\tilde{W}_{i j}^{-}$    (22)

$\widetilde{W}_{i j}^{+}$ and $\widetilde{W}_{i j}^{-}$ are overall assessment value of beneficial and non-beneficial criteria respectively.

Step 6: Allocate ranking to all the computed closeness values in descending order. In which the best alternative refer by higher closeness value that indicates the best performance, and vice versa.

4.1 FUZZY-MOORA based optimum parameters

In the present research work, the best parametric combination of input variables was determined during the machining of CP-Ti grade 2 using fuzzy embedded MOORA method. The objective was to minimize the tool wear, cutting force, and surface roughness. The interaction effects between the complexness of machining characteristics and process parameters create difficulties to recommend the best parametric combinations during machining.

When the decision-maker faces troubles to express quantitative values and dealing with situations that are too ill-defined and complex, in such conditions, decision-maker uses the non-numerical form in words known as linguistic terms such as excellent, good, very good, low, very low, poor etc., to express values in the qualitative thoughts of the prepared workpiece [24]. Therefore, selecting the optimum among the available parameters is a challenging task. Assessment of machining parameters of all alternatives i.e. very poor, poor, average, very good etc. with their triangular fuzzy numbers are shown in Table 3. The aforesaid linguistic variables are represented by triangular fuzzy numbers. The first linguistic variable (very very low) having TFN 0,0,0.1 whereas the last linguistic variables (very very high) of Table 3 having TFN (0.9, 1.0, 1.0), Therefore, the assigned triangular fuzzy numbers lie between 0 to 1.

Furthermore, mentioned linguistics variables and triangular fuzzy numbers are used to express the relative weight of each machining response. The specified linguistic variables were obtained by the relative weight of selected output responses such as flank wear, cutting force, and surface roughness as shown in Table 4. Tool wear is an unavoidable phenomenon during machining and it affects the cutting force and surface roughness of the workpiece. Therefore, relative weight for tool wear is kept at very very high (VVH) priority in Table 4 as compared to the cutting force and surface roughness. Whereas, cutting force and surface roughness are also essential criteria during machining are given relative weight very high (VH). The corresponding triangular fuzzy number values of criteria (Table 4) are referred from linguistic variables values of Table 3.

Furthermore, all the available alternatives were evaluated and validated based on linguistic variables. The values of triangular fuzzy nubers for linguistic variables are lied between 0 to 10. The seven different fuzzy linguistics variables such as very very good (VVG), very good (VG) good (G) fair (F), poor (P), very poor (VP) and very very poor (VVP) were obtained during the valuation as shown in Table 5.

Table 3. Linguistic variables used for each criterion

Table 4. Relative weights of each criterion

Table 5. Linguistic variables used for each alternative

Table 6. Results of the assessment

Furthrmore, aforementioned fuzzy linguistics variables have assigned to 27 alternatives to achive the best combination (refer Table 6). All crips values of output responses (F_c,VB_c and R_a) converted into the linguistic variables according to the prirority. For example, the range of lowest tool wear (0.082 mm) has denoted by very very good (VVG) and higher tool wear (0.277-0.295 mm) have denoted by very very poor (VVP). The assessment of the results is shown in Table 6. The output responses viz. flank wear, cutting force, and surface roughness are represented into fuzzy linguistics variables. After this, suitable triangular fuzzy numbers were prepared by the formation of a fuzzy decision matrix that is done by converting the data sets obtained after assessment.

Table 7. Fuzzy decision matrix

Table 8. Normalized fuzzy decision matrix

Table 9. Weighted normalized fuzzy decision matrix

The triangular fuzzy numbers of fuzzy decision matrix shown in Table 7 are obtained after conversion of linguistics variables of cutting force (F_c), tool wear (VB_c), and surface roughness (R_a) of Table 6. For example, the linguistic variable very very good (VVG) (refer to Table 5) having a triangular fuzzy number (9.10,10). Therefore, all places in Table 6 the VVG is replaced by (9,10,10) in Table 7.

The fuzzy decision matrix as shown in Table 7 was implemented using Eqns. (16-18) and the results are shown in Table 8 as a normalized fuzzy decision matrix. The fuzzy decision matrix has obtained by dividing all values of Table 7 by ten (10). Afterward, Table 9 expresses a weighted normalized fuzzy decision matrix, in which the relevant weight of every machining criterion was multiplied with their adjacent values. For example, the relative weight of flank wear is very very high (VVH) whereas the relative weights of cutting force and surface roughness are very high (VH) (refer Table 3). Therefore, adjacent values of relative weights multiplied by values of normalized fuzzy decision matrix values. After that applying Eq. (22) the listed values of Table 9 further converted into crisps values are represented in Table 10.

At last, Table 11 has been developed using Eq. (23) in which complete assessment of values. For example, assessment value (y_i) of alternative 1, (row 1) is calculated by the addition of consecutive crips values (refer Table 10) of cutting force (F_c), flank wear (VB_c), and surface roughness (R_a). The overall assessment values have been shown in decreasing order according to the preference ranking. The highest assessment values would be denoted by rank 1 whereas the lowest assessment value would be denoted by rank 27 because total twenty-seven experiments were conducted to obtain the best parametric combination.

The experiment number 24 has been observed the best operating parameter that gives optimum responses showing less flank wear, cutting force, and surface roughness. Whereas experiment number 11 has shown the worst responses, in which flank wear, cutting force, and surface roughness has the highest values.

Table 10. Crisp values for weighted normalized fuzzy decision matrix

Table 11. Overall assessment value

Therefore, results show that at higher cutting speed (140 m/min) and higher tool overhang length (65mm) with medium hardness (1934 HV) level, the tool wear recorded less with less cutting force and good surface roughness.

The aforementioned combination reported suitable and optimum for the machining, out of 27 experiments. With the increase in the cutting speed during machining of titanium alloys the cutting temperature also rises at shear zone due to friction during turning. The high temperature at the tool-workpiece interface remains high enough to influence the surface roughness of the workpiece. This high temperature allows regenerating the workpiece surface due to thermal expansion that results in the disappearance of micro cracks and cavities from the surface of the workpiece [25].

Experiment number 24 also depicted that increase in average microhardness reduces the cutting force and flank wear significantly, however, a high level of tool overhang length does not impact surface roughness and flank wear significantly. Therefore, a higher range of machining parameters considered in this study are fairly recommended for the machining of CP-Ti grade to titanium alloy.

4.2 Analysis of results

The main effect plots for cutting force, flank wear, and surface roughness are illustrated in Figures 5, 6, and 7 respectively. In Figure 5, It is depicted that cutting speed and tool overhang significantly affect the cutting force and its least value is obtained at the middle level of microhardness.

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Figure 5. Main effect plot for cutting force (Fc)

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Figure 6. Main effect plot for tool flank wear (VB_c)

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Figure 7. Main effect plot for Surface roughness (Ra)

From Figure 6, it is observed that microhardness and cutting speed play a significant role in flank wear because higher the cutting speed results in higher tool deformation and cutting temperature. However, an increase in microhardness results in an increase in wear resistance and cutting edge stability.

Figure 7 shows the main effect plot for surface roughness, it is observed that cutting speed, microhardness, and tool overhang are significantly affecting the surface roughness. At lower cutting speed, surface roughness is higher, whereas at higher cutting speed, the surface roughness is observed lower due to regeneration of the workpiece surface at a higher cutting temperature [8]. Whereas medium level of hardness and tool overhang reduces the vibrations and chattering of the cutting tool due to which lower surface roughness is obtained.