Multi-response optimization of Ti-6Al-4V turning operations using Taguchi-based grey relational analysis coupled with kernel principal component analysis
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Abstract
Ti-6Al-4V has a wide range of applications, especially in the aerospace field; however, it is a difficult-to-cut material. In order to achieve sustainable machining of Ti-6Al-4V, multiple objectives considering not only economic and technical requirements but also the environmental requirement need to be optimized simultaneously. In this work, the optimization design of process parameters such as type of inserts, feed rate, and depth of cut for Ti-6Al-4V turning under dry condition was investigated experimentally. The major performance indexes chosen to evaluate this sustainable process were radial thrust, cutting power, and coefficient of friction at the tool-chip interface. Considering the nonlinearity between the various objectives, grey relational analysis (GRA) was first performed to transform these indexes into the corresponding grey relational coefficients, and then kernel principal component analysis (KPCA) was applied to extract the kernel principal components and determine the corresponding weights which showed their relative importance. Eventually, kernel grey relational grade (KGRG) was proposed as the optimization criterion to identify the optimal combination of process parameters. The results of the range analysis show that the depth of cut has the most significant effect, followed by the feed rate and type of inserts. Confirmation tests clearly show that the modified method combining GRA with KPCA outperforms the traditional GRA method with equal weights and the hybrid method based on GRA and PCA.
Keywords
Ti-6Al-4V Taguchi method Grey relational analysis (GRA) Kernel principal component analysis (KPCA) Multi-response optimization1 Introduction
Because of their many excellent properties, such as high strength-to-weight ratio and good heat and corrosion resistance, titanium alloys have a wide range of applications, especially in the aerospace field with about 50% of the world’s total titanium [1]. Among them, one of the most commonly used is Ti-6Al-4V. However, this alloy is difficult to cut because of its low thermal conductivity and high chemical reactivity, which leads to rapid tool wear or breakage. About 80% of the heat generated during the cutting process is conducted into the cutting tool, which greatly weakens its cutting performance [2]. In order to improve the poor machinability of Ti-6Al-4V and reduce the machining cost, one way is to develop new tool materials to replace the commonly used cemented carbide [3], such as polycrystalline diamond (PCD) [4] and polycrystalline cubic boron nitride (PCBN) [5], which can improve the machinability of Ti-6Al-4V effectively. However, it is too expensive to be considered. Another way is to improve the tool structure, such as introducing friction-reducing grooves [6] or surface textures [7, 8] on the tool face, which can reduce cutting forces, decrease cutting temperature, and improve the friction condition at the tool-chip interface, thus enhancing tool life. However, surface texturing is often fabricated on inserts with a flat rake face or flank face, which is not commonly used. Research on the application of surface texturing to improve cutting performance of inserts with 3D complex topography has rarely been reported. Therefore, it is necessary to apply surface texturing to inserts with a 3D shape on the rake face to improve the machinability of Ti-6Al-4V. The most effective and advisable way is to optimize the existing process parameters, which requires lower investment and has better social sustainability without making drastic changes. In addition, it is most likely to be accepted by users [9, 10]. In the present work, the process parameters, such as type of inserts (with texture or not), feed rate, and depth of cut, were optimized to improve the machinability of Ti-6Al-4V.
In the machining process, common performance indexes for evaluating machinability include productivity, operating cost, and product quality [11]. Because of the inconvenience of direct measurement, they are usually described by measurable process outputs, such as tool life, surface roughness, cutting forces, power consumption, and vibration. Today, many studies focus on the optimization of individual performance characteristic in the machining processes. Subramanian et al. [12] developed a statistical model to envisage vibration amplitude in terms of process parameters such as radial rake angle, tool nose radius, cutting speed, feed rate, and axial depth of cut by response surface methodology (RSM) during end milling of Al 7075-T6. They found that the vibration amplitude was maintained at the minimum under the conditions of a radial rake angle of 12°, nose radius of 0.8 mm, cutting speed of 115 m/min, feed rate of 0.16 mm/r, and axial depth of cut of 2.5 mm. By combining genetic algorithms (GA) and RSM, Singh and Rao [13] obtained the optimum machining conditions of a cutting speed of 200 m/min, feed rate of 0.1 mm/r, effective rake angle of 6°, and nose radius of 1.2 mm for a minimum surface roughness value of 0.486 6 µm during hard turning of AISI 52100 steel. Ramana et al. [14] concluded that the optimal process parameters for minimizing tool wear were an MQL machining environment, cutting speed of 63 m/min, feed rate of 0.274 mm/r, depth of cut of 1.0 mm, and uncoated tool during turning of Ti-6Al-4V by Taguchi’s robust design methodology. In short, all of these studies have achieved good results without considering other indexes. However, in practice, the optimization of the individual performance index is often achieved at the cost of deteriorating other indexes due to the complexity and conflict between the indexes. For example, a higher cutting speed helps increase productivity and surface quality, but also accelerates tool wear, leading to shorter tool life and increased production costs. Therefore, in order to achieve sustainable manufacturing, especially for titanium alloys, it is more reasonable to optimize multiple responses simultaneously based on environmentally sustainable, economic, and technical requirements.
Related research on multi-response optimization in the cutting process, considering several contradictory responses, has also been frequently reported. Yan and Li [15] optimized cutting parameters such as spindle speed, feed rate, depth of cut, and width of cut in the milling process based on weighted grey relational analysis (GRA) and RSM in order to evaluate trade-offs between sustainability, production rate, and cutting quality. Surface roughness, material removal rate, and cutting energy were evaluated simultaneously. Sarıkaya and Güllü [16] optimized the process parameters such as cutting fluid, fluid flow rate, and cutting speed by Taguchi based GRA, taking flank wear, notch wear, and surface roughness as process performance indexes. In order to minimize surface roughness and burr formation concurrently, a multi-objective particle swarm optimization method was utilized to find the optimal cutting parameters during micro-end milling of Ti-6Al-4V by Thepsonthi and Özel [17]. Yi et al. [18] presented a multi-objective optimization model based on a non-dominated sorting GA to explore the impact of cutting speed and feed rate on carbon emissions and processing time. To obtain favorable performance characteristics such as material removal rate, cutting force, and surface roughness during dry turning of AISI 304 austenitic stainless steel, Nayak et al. [19] combined Taguchi method and GRA to optimize the machining parameters such as cutting speed, feed rate, and depth of cut. Mia et al. [20] first attempted to optimize cutting forces, surface roughness, cutting temperature, and chip reduction coefficient during turning of Ti-6Al-4V under dry and high-pressure coolant using GRA integrated with Taguchi method. Afterwards, they experimentally investigated surface roughness, cutting force, and feed force during turning of Ti-6Al-4V under cryogenic (liquid nitrogen) condition and performed the desirability-based multi-response optimization according to the models of responses by RSM and artificial neural network [21].
All the above literatures have achieved the multi-objective optimization that simultaneously meets the environment sustainable, economic, and technical requirements by different indexes and methods. At the same time, it also shows the complexity and diversity of indexes. In the present work, performance indexes such as radial thrust, cutting power, and coefficient of friction at the tool-chip interface were chosen to evaluate the machinability of Ti-6Al-4V for the following reasons. Firstly, in spite of not doing any work, radial thrust may lead to deformation or vibration of the workpiece, which will greatly increase the likelihood of tool breakage and directly affect machining accuracy and surface quality, thereby indirectly increasing the processing costs [22]. Therefore, it can be taken as a technical or economic index. Besides, research on the use of radial thrust as an optimization index has rarely been reported. Secondly, the contradiction between the growing demand for energy and the increasing severity of resource shortage, along with more and more serious environmental problems, forces us to pay more attention to energy conservation and emissions reduction, especially in manufacturing. Energy savings can reach up to 6%–40% by selecting the optimal combination of process parameters [23]. Power consumption is a very important index reflecting energy consumption during the cutting process, which can be derived from the measured tangential force and axial thrust. Because more difficult-to-cut materials will require more power and greater cutting force or torque [11], power consumption can be considered as an environment sustainable or technical index. Thirdly, coefficient of friction at the tool-chip interface is influenced by many factors such as work material, tool material, tool geometry, cutting parameters, and cutting fluid. However, it has a great influence on many other indexes such as energy consumption and tool life. The larger the coefficient of friction at the tool-chip interface is, the more energy will be consumed and the more severe tool wear will appear. Hence, it can be regarded as an environment sustainable or economic index. Furthermore, little research has been reported about taking coefficient of friction at the tool-chip interface as an optimization index for Ti-6Al-4V turning according to the previous researches. The only relevant study by Mia et al. [24] was reported not long ago, which focused on the optimization of chip compression ratio, effective shear angle, friction coefficient at the tool rake surface and the chip-tool interface temperature in turning process using grey relation-based Taguchi method. At last, these three indexes are easy to obtain based on the acquired cutting force signals.
Based on the implementation method of the optimization procedure, the multi-objective optimization method can be divided into the priori [15, 16, 19] and posterior technique [17, 18]. Due to the advantages of small number of tests, simple calculations, and unique optimal results, the priori technique like Taguchi-based GRA has always been favored by many scholars, whose basic idea is to transform the multi-objective optimization problem into a single objective optimization problem [25]. However, a key question exists that the weights of multiple performance indexes on the single objective are hard to determine. In many literatures, equal weights [16, 26] or weights assigned by decision makers [2, 27] are often applied to describe the relative importance of each performance index, which results in great uncertainty and irrationality. In order to solve this problem, many scholars have devoted much attention and made some achievements. Lu et al. [28] performed the principal component analysis (PCA) on the grey relational coefficients (GRC) corresponding to various performance indexes, and extracted the square values of three components in the first principal component as the weighting values. Dubey and Yadava [29] applied a hybrid approach based on the Taguchi method and PCA for multi-objective optimization by extracting three principal components from the three quality characteristics, and then transforming into a unique total principal component index with the proportion of each eigenvalue as weighting value. It is well known that the basic principle of PCA is to apply the linear combinations of the original correlation variables to represent the main characteristics of the objectives, which is a method to solve multicollinearity. However, the relationship between various performance indexes in the cutting process tends to be nonlinear, so it is no longer appropriate to use PCA to deal with this multi-response optimization problem. Because kernel principal component analysis (KPCA) is a nonlinear extension of PCA using kernel technique, KPCA can be applied to solve nonlinear problems in multi-response optimization.
2 Methodology and experimental design
2.1 Taguchi method
2.2 Principle of GRA
Grey system theory is a systematic scientific theory proposed by Deng in Ref. [31] to solve the problems with insufficient, poor, or uncertain information. Actually, the optimization problem of multiple performance indexes with the lower-the-better characteristics itself contains a certain degree of uncertainty and ambiguity. Therefore, the grey system theory is very suitable for solving this multi-response optimization problem. Based on this theory, GRA is carried out to evaluate the degree of correlation between the observed data (comparability sequence) and the desired value (reference sequence), and transform multiple performance characteristics into a single grey relational grade (GRG) value as the optimization criterion.
However, due to the nonlinearity between the performance indexes, this optimization criterion is not very reasonable. In order to reveal the principal features of these performance indexes, KPCA will be introduced and an improved \(\gamma\) will be proposed, with the corresponding weights also obtained in the next section.
2.3 Principle of KPCA
- (i)Constituting the original performance index array \(\xi_{i} (k), i = \, 1, \, 2, \cdots ,m;k = \, 1, \, 2, \cdots ,s\)where m and s are the number of experiments and performance indexes, respectively.$${\varvec{\varXi}} = \left[ {\begin{array}{*{20}c} {\xi_{1} (1)} & {\xi_{1} (2)} & \cdots & {\xi_{1} (s)} \\ {\xi_{2} (1)} & {\xi_{2} (2)} & \cdots & {\xi_{2} (s)} \\ \vdots & \vdots & \vdots & \vdots \\ {\xi_{m} (1)} & {\xi_{m} (2)} & \cdots & {\xi_{m} (s)} \\ \end{array} } \right],$$(5)
- (ii)Selecting the kernel function \(\psi\) and calculating the kernel matrix Kwhere \({\varvec{\varXi}}_{\mu }\) and \({\varvec{\varXi}}_{\nu }\) denote the performance index vector at the μth and vth experiment; \(\mu = 1,2, \cdots ,m;v = 1,2, \cdots ,m\) . In this paper, Gaussian radial basis function is selected as the kernel function.$${K}_{\mu \nu } = \varvec{K}({\varvec{\varXi}}_{\mu } ,{\varvec{\varXi}}_{\nu } ) = \left\langle {\psi ({\varvec{\varXi}}_{\mu } ),\psi ({\varvec{\varXi}}_{\nu } )} \right\rangle = \exp \left( { - {{\left\| {{\varvec{\varXi}}_{\mu } - {\varvec{\varXi}}_{\nu } } \right\|^{2} } \mathord{\left/ {\vphantom {{\left\| {{\varvec{\Xi}}_{\mu } - {\varvec{\varXi}}_{\nu } } \right\|^{2} } {\left( {2\sigma^{2} } \right)}}} \right. \kern-0pt} {\left( {2\sigma^{2} } \right)}}} \right),$$(6)
- (iii)Centering the kernel matrix K$${K}_{\mu \nu }^{*} = K_{\mu \nu } - \frac{1}{m}\left( {\sum\limits_{\omega = 1}^{m} {K_{\mu \omega } + \sum\limits_{\tau = 1}^{m} {K_{\tau \nu } } } } \right) + \frac{1}{{m^{2} }}\sum\limits_{\omega ,\tau = 1}^{m} {K_{\omega \tau } } .$$(7)
- (iv)Determining the eigenvalues and eigenvectors from the above centered matrix K^{*}where \(\lambda_{i}^{*}\) is the ith eigenvalue, \(i = 1,2, \cdots ,m;\)\({{\varvec{\alpha}}_{i}} = \left( {\alpha_{i1} ,\alpha_{i2} , \cdots ,\alpha_{im} } \right)^{\text{T}}\) is the corresponding eigenvector. Assuming that \(\lambda_{1}^{*} \geqslant \lambda_{2}^{*} \geqslant\cdots \geqslant \lambda_{m}^{*}\), and the eigenvectors are normalized.$${\lambda_{i}^{*}} {{\varvec{\alpha}}_{i}} = {\varvec{K}}^{{\mathbf{*}}} {{\varvec{\alpha}}_{i}} ,$$(8)
- (v)Calculating the contribution rate \(c(k)\) of the first s eigenvalues as the weight of the performance index and extracting the corresponding kernel principal components \(p_{i} (k)\)$$c(k) = {{\lambda_{k}^{*} } \mathord{\left/ {\vphantom {{\lambda_{k}^{*} } {\sum\limits_{i = 1}^{m} {\lambda_{i}^{*} } }}} \right. \kern-0pt} {\sum\limits_{i = 1}^{m} {\lambda_{i}^{*} } }},$$(9)$$p_{i} (k) = \sum\limits_{j = 1}^{s} {\xi_{i} } (j)\,\alpha_{kj} .$$(10)
All the eigenvalues are arranged in descending order with respect to variance. Usually, the accumulative contribution rate of the first three eigenvalues reaches over 90%, which means the first three kernel principal components hold the most amount of information in the data. Therefore, the first three eigenvalues are taken into consideration in this work.
Based on this optimization criterion, the effect of each factor at different levels can be evaluated, and the optimum parameter combination corresponding to the maximum \(\gamma_{k}\) is obtained, which can be compared with that obtained from \(\gamma\).
2.4 Design of experiment and test results
Process parameters and their levels in the tests
Process parameter | Symbol | Level 1 | Level 2 | Level 3 | Level 4 |
---|---|---|---|---|---|
Type of inserts | A | Nose 1 | Nose 2^{*} | ||
Feed rate/ (\(\text{mm}\cdot\text{r}^{ - 1}\)) | B | 0.10 | 0.15^{*} | 0.20 | 0.25 |
Depth of cut/ mm | C | 0.5 | 1.0^{*} | 1.5 | 2.0 |
Experimental layout using L_{16} (2^{1}×4^{2}) and their responses
Run No. | Factors and their levels | F_{y} /N | P/W | μ | ||
---|---|---|---|---|---|---|
A | B | C | ||||
1 | 1 | 1 | 1 | 160.09 | 85.80 | 2.20 |
2 | 1 | 2 | 2 | 225.33 | 208.49 | 2.12 |
3 | 1 | 3 | 3 | 245.72 | 409.87 | 1.92 |
4 | 1 | 4 | 4 | 274.12 | 577.59 | 2.03 |
5 | 1 | 1 | 2 | 177.10 | 200.72 | 1.75 |
6 | 1 | 2 | 1 | 188.69 | 101.66 | 2.58 |
7 | 1 | 3 | 4 | 253.26 | 553.70 | 1.84 |
8 | 1 | 4 | 3 | 266.04 | 428.02 | 2.15 |
9 | 2 | 1 | 3 | 169.67 | 341.47 | 1.46 |
10 | 2 | 2 | 4 | 229.23 | 507.53 | 1.64 |
11 | 2 | 3 | 1 | 200.63 | 107.58 | 2.89 |
12 | 2 | 4 | 2 | 245.34 | 222.39 | 2.97 |
13 | 2 | 1 | 4 | 181.15 | 481.27 | 1.37 |
14 | 2 | 2 | 3 | 206.84 | 358.48 | 1.81 |
15 | 2 | 3 | 2 | 223.52 | 215.21 | 2.52 |
16 | 2 | 4 | 1 | 193.54 | 101.12 | 3.43 |
3 Results and discussion
In this section, the approach based on GRA and KPCA to optimize the process parameters for Ti-6Al-4V turning is presented in detail, and the multi-response optimization results by \(\gamma\), the integrated optimization criterion \(\gamma_{\text{p}}\), and \(\gamma_{k}\) are compared and validated through the confirmation tests. Here, \(\gamma_{\text{p}}\) denotes the corresponding grey relational grade obtained from the hybrid method based on GRA and PCA, which is used to illustrate the effect of nonlinearity.
3.1 Detailed optimization procedure
S/N and its normalization for radial thrust force, cutting power and coefficient of friction
Run No. | S/N ratios for different responses | Normalization for different responses | ||||
---|---|---|---|---|---|---|
F _{ y} | P | μ | F _{ y} | P | μ | |
1 | −44.09 | −39.67 | −6.85 | 1.000 0 | 1.000 0 | 0.483 5 |
2 | −47.06 | −46.38 | −6.52 | 0.364 5 | 0.534 4 | 0.525 0 |
3 | −47.81 | −52.25 | −5.67 | 0.203 4 | 0.179 9 | 0.632 0 |
4 | −48.76 | −55.23 | −6.13 | 0.000 0 | 0.000 0 | 0.574 5 |
5 | −44.96 | −46.05 | −4.84 | 0.812 3 | 0.554 3 | 0.736 1 |
6 | −45.52 | −40.14 | −8.24 | 0.694 4 | 0.911 0 | 0.309 5 |
7 | −48.07 | −54.87 | −5.28 | 0.147 2 | 0.022 2 | 0.681 6 |
8 | −48.50 | −52.63 | −6.64 | 0.055 7 | 0.157 2 | 0.510 3 |
9 | −44.59 | −50.67 | −3.30 | 0.892 0 | 0.275 6 | 0.930 6 |
10 | −47.21 | −54.11 | −4.28 | 0.332 5 | 0.067 8 | 0.807 4 |
11 | −46.05 | −40.63 | −9.20 | 0.580 3 | 0.881 3 | 0.188 3 |
12 | −47.80 | −46.94 | −9.46 | 0.206 3 | 0.500 5 | 0.156 5 |
13 | −45.16 | −53.65 | −2.74 | 0.770 3 | 0.095 7 | 1.000 0 |
14 | −46.31 | −51.09 | −5.14 | 0.523 7 | 0.250 2 | 0.698 5 |
15 | −46.99 | −46.66 | −8.05 | 0.379 5 | 0.517 7 | 0.333 8 |
16 | −45.74 | −40.10 | −10.07 | 0.647 2 | 0.913 8 | 0.000 0 |
Reference sequence | 1.000 0 | 1.000 0 | 1.000 0 |
Calculated Δ_{oi}, \(\xi\), \(\gamma\), and the corresponding ranking of \(\gamma\)
Run No. | Deviation sequences | Grey relational coefficient | \(\gamma\) | Ranking | ||||
---|---|---|---|---|---|---|---|---|
Δ_{oi}(F_{y}) | Δ_{oi}(P) | Δ_{oi}(μ) | \(\xi \left( {F_{y} } \right)\) | \(\xi \left( P \right)\) | \(\xi \left( \mu \right)\) | |||
1 | 0.000 0 | 0.000 0 | 0.516 5 | 1.000 0 | 1.000 0 | 0.491 9 | 0.830 6 | 1 |
2 | 0.635 5 | 0.465 6 | 0.475 0 | 0.440 3 | 0.517 8 | 0.512 8 | 0.490 3 | 10 |
3 | 0.796 6 | 0.820 1 | 0.368 0 | 0.385 6 | 0.378 8 | 0.576 0 | 0.446 8 | 12 |
4 | 1.000 0 | 1.000 0 | 0.425 5 | 0.333 3 | 0.333 3 | 0.540 3 | 0.402 3 | 16 |
5 | 0.187 7 | 0.445 7 | 0.263 9 | 0.727 1 | 0.528 7 | 0.654 5 | 0.636 8 | 4 |
6 | 0.305 6 | 0.089 0 | 0.690 5 | 0.620 6 | 0.848 9 | 0.420 0 | 0.629 9 | 5 |
7 | 0.852 8 | 0.977 8 | 0.318 4 | 0.369 6 | 0.338 3 | 0.610 9 | 0.439 6 | 13 |
8 | 0.944 3 | 0.842 8 | 0.489 7 | 0.346 2 | 0.372 3 | 0.505 2 | 0.407 9 | 15 |
9 | 0.108 0 | 0.724 4 | 0.069 4 | 0.822 4 | 0.408 4 | 0.878 2 | 0.703 0 | 2 |
10 | 0.667 5 | 0.932 2 | 0.192 6 | 0.428 3 | 0.349 1 | 0.721 9 | 0.499 8 | 9 |
11 | 0.419 7 | 0.118 7 | 0.811 7 | 0.543 7 | 0.808 2 | 0.381 2 | 0.577 7 | 7 |
12 | 0.793 7 | 0.499 5 | 0.843 5 | 0.386 5 | 0.500 3 | 0.372 2 | 0.419 6 | 14 |
13 | 0.229 7 | 0.904 3 | 0.000 0 | 0.685 2 | 0.356 0 | 1.000 0 | 0.680 4 | 3 |
14 | 0.476 3 | 0.749 8 | 0.301 5 | 0.512 1 | 0.400 0 | 0.623 8 | 0.512 0 | 8 |
15 | 0.620 5 | 0.482 3 | 0.666 2 | 0.446 2 | 0.509 0 | 0.428 7 | 0.461 3 | 11 |
16 | 0.352 8 | 0.086 2 | 1.000 0 | 0.586 3 | 0.853 0 | 0.333 3 | 0.590 9 | 6 |
The first three kernel principal components, \(\gamma_{k}\) and the corresponding ranking of \(\gamma_{k}\)
Run No. | Kernel principal components | \(\gamma_{k}\) | Ranking | ||
---|---|---|---|---|---|
First | Second | Third | |||
1 | 0.285 3 | − 0.092 4 | − 0.008 8 | 0.206 3 | 1 |
2 | − 0.018 1 | 0.042 8 | 0.001 6 | 0.027 2 | 16 |
3 | − 0.094 4 | 0.036 1 | − 0.001 1 | 0.070 3 | 11 |
4 | − 0.114 7 | 0.057 9 | − 0.003 4 | 0.090 8 | 6 |
5 | 0.021 4 | − 0.074 7 | − 0.002 8 | 0.041 4 | 14 |
6 | 0.169 7 | 0.023 4 | 0.005 2 | 0.110 7 | 4 |
7 | − 0.118 6 | 0.029 3 | − 0.001 3 | 0.082 2 | 8 |
8 | − 0.091 6 | 0.065 4 | − 0.003 0 | 0.079 8 | 10 |
9 | − 0.044 1 | − 0.164 2 | − 0.001 1 | 0.089 1 | 7 |
10 | − 0.122 5 | − 0.017 0 | 0.002 2 | 0.079 9 | 9 |
11 | 0.144 1 | 0.053 7 | 0.006 0 | 0.106 9 | 5 |
12 | − 0.010 0 | 0.093 6 | − 0.002 2 | 0.041 7 | 13 |
13 | − 0.111 1 | − 0.161 9 | 0.007 2 | 0.128 4 | 2 |
14 | − 0.069 6 | − 0.010 2 | − 0.001 5 | 0.045 6 | 12 |
15 | − 0.004 7 | 0.063 8 | − 0.001 7 | 0.027 2 | 15 |
16 | 0.178 8 | 0.054 5 | 0.004 7 | 0.128 0 | 3 |
The last columns of Tables 4 and 5 show the rankings of \(\gamma\) and \(\gamma_{k}\), respectively. According to the previous analysis, a larger \(\gamma\) or \(\gamma_{k}\) means that the observed sequence is more closely related to the reference sequence representing the best performance. Thus, the largest \(\gamma\) and the largest \(\gamma_{k}\) correspond to the optimal parameter combinations, respectively. By comparing the ranking results of the last column in Tables 4 and 5, it can be found that among all the tests, the largest \(\gamma\) and the largest \(\gamma_{k}\) both appear under Run No.1, which means that in the 16 tests studied, the optimal combinations of process parameters determined based on these two methods are consistent. However, in order to find out the relative importance of the process parameters on the integrated optimization criterion, it is necessary to further analyze the effect of each parameter level to identify the optimal combination of process parameters more accurately.
3.2 Effect of process parameters on the integrated optimization criterion
Response table for γ
Experimental factors | Average γ | Range | Ranking | |||
---|---|---|---|---|---|---|
Level 1 | Level 2 | Level 3 | Level 4 | |||
Type of inserts A | 0.535 5 | 0.555 6^{*} | 0.020 1 | 3 | ||
Feed rate B | 0.712 7^{*} | 0.533 0 | 0.481 4 | 0.455 2 | 0.257 5 | 1 |
Depth of cut C | 0.657 3^{*} | 0.502 0 | 0.517 4 | 0.505 5 | 0.155 3 | 2 |
Response table for \(\gamma_{k}\)
Experimental factors | Average \(\gamma_{k}\) | Range | Ranking | |||
---|---|---|---|---|---|---|
Level 1 | Level 2 | Level 3 | Level 4 | |||
Type of inserts A | 0.088 6^{*} | 0.080 8 | 0.007 7 | 3 | ||
Feed rate B | 0.116 3^{*} | 0.065 8 | 0.071 7 | 0.085 1 | 0.050 5 | 2 |
Depth of cut C | 0.137 9^{*} | 0.034 4 | 0.071 2 | 0.095 3 | 0.103 6 | 1 |
However, in terms of \(\gamma_{k}\) in Table 7, the combination of process parameters A_{1}B_{1}C_{1} (type of inserts: Nose 1, feed rate: 0.1 mm/r, and depth of cut: 0.5 mm), has the best performance with the maximum average responses. Furthermore, the results of range analysis reveal that depth of cut is the most significant factor that affects the integrated optimization criterion \(\gamma_{k}\).The detailed effects of the process parameters are illustrated in Fig. 5b. Compared with the conclusions drawn from Fig. 5a, it can be inferred that in order to obtain the maximum integrated optimization criterion, Fig. 5b gives a similar opinion in the aspect of depth of cut and feed rate, but different in type of inserts. This may be caused by the nonlinear relationship between the performance indexes. In Fig. 5a, there is an approximate linear relationship, while in Fig. 5b, the nonlinear relationship can be clearly observed. Therefore, the effect of nonlinearity on multi-response optimization results deserves further study.
3.3 Effect of nonlinearity on multi-response optimization results
Eigenvalues and the corresponding contribution rate for principal components
Principal components | Eigenvalues | Contribution rate/% |
---|---|---|
First | 0.071 7 | 60.76 |
Second | 0.044 7 | 37.83 |
Third | 0.001 7 | 1.41 |
Eigenvectors for each principal component
Performance index | Eigenvectors | ||
---|---|---|---|
First principal component | Second principal component | Third principal component | |
Radial thrust | 0.443 8 | 0.696 5 | −0.563 8 |
Cutting power | 0.814 4 | −0.051 0 | 0.578 0 |
Coefficient of friction | −0.373 8 | 0.715 7 | 0.589 9 |
Contribution rate of each performance index to the fist principal component
Performance index | Contribution rate/% | Weight |
---|---|---|
Radial thrust | 19.70 | 0.197 0 |
Cutting power | 66.33 | 0.663 3 |
Coefficient of friction | 13.97 | 0.139 7 |
Response table for \(\gamma_{\text{p}}\)
Experimental factors | Average \(\gamma_{\text{p}}\) | Range | Rank | |||
---|---|---|---|---|---|---|
Level 1 | Level 2 | Level 3 | Level 4 | |||
Type of inserts A | 0.537 3 | 0.538 3^{*} | 0.001 0 | 3 | ||
Feed rate B | 0.645 2^{*} | 0.529 0 | 0.493 0 | 0.484 0 | 0.161 2 | 2 |
Depth of cut C | 0.774 3^{*} | 0.508 1 | 0.450 6 | 0.418 1 | 0.356 2 | 1 |
In order to demonstrate the effectiveness of the proposed method, the obtained optimization results need further verification. The verification tests and comparison results were presented in Sect. 3.4.
3.4 Verification tests
Comparison between the initial combination and the optimal combinations
Combinations | Symbol | Radial thrust | Power consumption | Coefficient of friction |
---|---|---|---|---|
Initial | A _{2} B _{2} C _{2} | 204.17 | 206.51 | 2.21 |
Optimized by \({\gamma \mathord{\left/ {\vphantom {\gamma {\gamma_{\text{p}} }}} \right. \kern-0pt} {\gamma_{\text{p}} }}\) | A _{2} B _{1} C _{1} | 163.12 | 88.03 | 2.17 |
Relative increment | −20.11% | −57.37% | −1.67% | |
Optimized by \(\gamma_{k}\) | A _{1} B _{1} C _{1} | 160.09 | 85.80 | 2.20 |
Relative increment | −21.59% | −58.45% | −0.25% |
It can be seen that relative to the initial condition, there is a clear improvement in terms of radial thrust and power consumption, and a minor improvement in the coefficient of friction for both of the optimization results. As illustrated in Table 12, under the optimal combination A_{1}B_{1}C_{1} obtained by \(\gamma_{k}\), the power consumption is reduced by 58.45%, and the radial thrust is decreased by 21.59%, which is superior to those under the optimal combination A_{2}B_{1}C_{1} obtained by \(\gamma\) and \(\gamma_{\text{p}}\), though there is little difference in the coefficient of friction. Consequently, the results of these tests reveal that the modified method combining GRA with KPCA is very effective for solving multi-response optimization problems and outperforms the single-GRA method with equal weights and the hybrid method based on GRA and PCA.
4 Conclusions
- (i)
KPCA was performed to extract the kernel principal components and identify the corresponding weights. The first three kernel principal components were extracted and the corresponding weights obtained from \(\zeta\) were 59.87%, 38.16%, and 1.98%, respectively.
- (ii)
Within the range of process parameters studied, the largest \(\gamma\) and \(\gamma_{k}\) both appeared in Run No. 1.
- (iii)
For \(\gamma\) and \(\gamma_{\text{p}}\), A_{2}B_{1}C_{1} is the optimal combination of process parameters, while for \(\gamma_{k}\), the combination A_{1}B_{1}C_{1} is the best.
- (iv)
Feed rate has the most dominant effect on \(\gamma\), followed by depth of cut and type of insert, while whereas for \(\gamma_{\text{p}}\) and \(\gamma_{k}\), the depth of cut is the most significant factor. A consistent view is that the optimal comprehensive performance can be obtained at a low feed rate and low depth of cut.
- (v)
The comparison results show that the nonlinearity in the multi-response optimization problem cannot be neglected.
- (vi)
Verification tests reveal that there is a clear improvement in terms of radial thrust and power consumption, and a minor improvement in the coefficient of friction at the tool-chip interface for both of the optimization results, but the proposed method outperforms the single-GRA method with equal weights and the hybrid method based on GRA and PCA.
Notes
Acknowledgements
The authors would like to acknowledge the financial assistance from the National Science and Technology Major Project of China (Grant No. 2012ZX04003-021).
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