# Energy Consumption Analysis Of Machining Centers Using Bayesian Analysis And
Genetic Optimization

Johnatan Cardona Jiménez  María I. Ardila  J. S. Rudas  Cesar A. Isaza M
Edwin J. Núñez  Miguel A. Rodriguez

###### Abstract

Responding to the current urgent need for low carbon emissions and high
efficiency in manufacturing processes, the relationships between three
different machining factors (depth of cut, feed rate, and spindle rate) on
power consumption and surface finish (roughness) were analysed by applying a
Bayesian seemingly unrelated regressions (SUR) model. For the analysis, an
optimization criterion was established and minimized by using an optimization
algorithm that combines evolutionary algorithm methods with a derivative-based
(quasi-Newton) method to find the optimal conditions for energy consumption
that obtains a good surface finish quality. A Bayesian ANOVA was also
performed to identify the most important factors in terms of variance
explanation of the observed outcomes. The data were obtained from a factorial
experimental design performed in two computerized numerical control (CNC)
vertical machining centers (Haas UMC-750 and Leadwell V-40iT). Some results
from this study show that the feed rate is the most influential factor in
power consumption, and the depth of cut is the factor with the stronger
influence on roughness values. An optimal operational point is found for the
three factors with a predictive error of less than 0.01% and 0.03% for the
Leadwell V-40iT machine and the Haas UMC-750 machine, respectively.

###### keywords:

Machining centres, Depth of cut, Feed rate, Spindle rate, Power consumption,
Roughness, Bayesian inference and Genetic optimization.

###### PACS:

0000 , 1111

###### MSC:

0000 , 1111

††journal: Nuclear Physics B

[inst1]organization=Faculty of Engineering, Pascual Bravo University,
city=Medellin, country=Colombia

[inst2]organization=Faculty of Engineering, Instituto Tecnologico
Metropolitano,city=Medellin, country=Colombia

##  1 Introduction

Energy-saving and emissions reduction are important challenges faced by
different economic sectors due to their negative potential impacts on the
environment and the acceleration of climate change. Improving energy
efficiency, which reduces energy consumption, is an important task for almost
all industrial processes, especially for the manufacturing industry. This
poses a challenging opportunity to mitigate the current levels of emissions
and reduce energy use through an appropriate assessment and characterization
of manufacturing industry operations for improving their performance [1].
According to the International Energy Agency (IEA), the manufacturing industry
is responsible for approximately 33% of global energy consumption and as much
as 36% of global carbon emissions [1, 2]. For machining processes (i.e.,
drilling, turning, or milling), a maximum efficiency of 30% has been reported
[2]. In general, the industrial sector is expected to gradually grow over the
next few decades, leading to increased demand for machining processes and,
consequently, increased energy consumption and carbon emissions [3]. Energy
savings for machining processes can be classified into two branches. The first
is the improvement of cutting technologies, such as high-speed, cutting tools
and their materials. The second is to examine the relationship between the
process parameters and the power required for process functionality to
optimize energy consumption by applying models and optimization methods for
energy savings. For instance, [4] proposed two alternatives to analyse the
optimization problem: design strategies and operational strategies. Design
strategies include all the aspects related to machine improvement, such as
power aspects and maintenance prediction, which includes the control systems
[5, 6]. For most common industries, design strategies are not the best
solution because they require downtime or machine center inactivity for
strategy implementation, which can increase economic losses. On the other
hand, operation strategies correspond to all the operational variables and
tools, including cutting parameters, tool paths, and process planning [7, 8,
9]. The most common factors considered in operational strategies are the feed
rate [10, 11, 12], depth of cut [10, 6, 1, 5, 13], cutting speed [14, 12, 15]
cutting edge radius [16, 17] and vibration signals [18, 15]. Operational
strategies have been focused on output variable optimization through
experimental, theoretical, or mixed approaches. Some of the most studied
output variables reported in the literature are the specific energy
coefficient or power machining consumption, the surface finish or surface
roughness, and the material removal rate [10, 6, 1, 5, 13, 14, 12, 19, 20].

Currently, engineers are aware that setting the correct operational variables
is the manufacturer’s main purpose. Thus, truly understanding the
interdependencies among the main operational factors and establishing the
correct levels for the inputs in the machining process is a technological
challenge [14, 21, 22]. In this sense, several authors have developed and
proposed different approaches for optimizing machining parameters. For
example, [14] developed a flank wear and surface roughness prediction model
for a hot turning machining process based on response surface methodology
(RSM). Their model had an 86.17% accuracy for surface roughneslinearities.
However, its prediction accuracy is not the best for all phenomena.
Considering a different approach, [18] used an artificial neural network (ANN)
for surface roughness prediction considering the cutting speed, feed rate, and
the depth of cut as input factors. The authors validated their hypothesis
about the strong relationship between the feed rate and the surface finish,
and they reported a maximum relative prediction error of 12.2% for the ANN.
[23] proposed an ANN for the prediction of surface roughness. Their
experiments were conducted on a CNC milling machine according to the
principles of Taguchi’s design of experiments, where the depth of cut, the
feed rate per tooth, the cutting speed, the engagement and the wear of the
cutting tool, the use of cutting fluid, and the three components of the
cutting force were the controlled parameters. The author found the most
influential parameters and attained a prediction level for the surface
roughness with a mean percentage squared error equal to 1.86%. In a similar
approach, [24] used a Taguchi optimization method for evaluating the cutting
speed, feed rate, and cut depth of cut in AISI H13 steel machined with a TiN-
coated P10 carbide tool. The authors used an orthogonal array, signal-to-noise
(S/N) ratio, and Pareto analysis of variance (ANOVA) to analyse the effect of
these milling parameters. They concluded that the Taguchi metho was viable for
solving machining problems with a minimum number of procedures. [25] used an
ANN predictive model for critical process parameters to predict the cutting
forces. For its implementation, the authors used a particle swarm optimization
(PSO) algorithm to optimize cutting conditions such as feed and speed. They
reported machining time reductions of up to 35%. [26] studied the cutting
forces, spindle motor current, and sound pressure level in a CNC milling
process to examine tool wear. The authors used a neural network-based sensor
fusion model and proposed newer methods such as feature space filtering and
prediction space filtering, among others, to improve prediction accuracy in a
real-time error-prone environment. [27] implemented another predicted cutting
force model using regression and an ANN. The authors used the methodology to
filter out abnormal data points, and the filtered data were used in a neural
network for better prediction.

During a machining process, the cutting time is composed of different stages:
the standby time (the whole system going towards the machining planning
setup), the air time (the tool does not have any applied load), and effective
cutting time (material removal through tool movement). A schematic diagram of
the power profile showing the cutting stages is illustrated in Figure 1. All
these cutting stages have a strong influence on the energy consumption for
this type of process. Similarly, several studies have shown that the toolpath,
cutting speed and depth, and spindle speed can strongly influence the
effective cutting time. The effect of these parameters on the total power
consumption of machining have also been noted [4, 22, 21]. Regardless of the
known relationship between the process parameters, the power consumption, and
the machining quality, it is necessary to extensively study, analyse, and
predict the quality of the surface finish of manufactured parts considering
optimal energy performance through the proper choice of operational factors.

Figure 1: Power profile of a machining process

This work presents a novel statistical analysis for an experimental study to
analyse the energy consumption (EC) and surface roughness (SR) and their
relationship with three controllable machining parameters. Specifically, in
this experiment, two ma chine centers (Haas UMC-750 and Leadwell V-40iT® CNC)
are tested under different experimental conditions to characterize their EC
and SR produced over experimental units both machines have processed. The
three machine parameters related to the experimental conditions are depth of
cut, feed rate, and spindle rate. Considering the plausible observed
correlation between the experimental outcomes (EC and SR), a bivariate
response regression linear model is used. A Bayesian approach is applied for
the inferential analysis of the latent quantities related to the statistical
modelling. An optimization criterion based on the joint minimization of both
outcomes is defined to find the optimal operational conditions for both
machining centers. In addition, a genetic algorithm is used to identify the
experimental conditions that jointly minimize the EC and SR for both machining
centers. To characterize the relevance of each controllable machining
parameter on both responses (EC and SR), a Bayesian version of the classic
analysis of variance (ANOVA) is used. The next section describes the
experimental setup employed in this study. It also presents the statistical
model and optimization procedures used to analyse the data obtained from the
experimental runs. Section 3 presents the main results and analysis from this
study. Finally, Section 4 provides some concluding remarks and potential open
problems for future development.

##  2 Materials and Methods

###  2.1 Machining centers and experimental design

The experimental runs are carried out on two CNC vertical machining centers: a
Haas UMC-750 with a 12000 rpm maximum speed and 22.4 kW of spindle power and a
Leadwell V-40iT with a 10000 rpm maximum speed and 18.5 kW of spindle power.
For a matter of simplicity, from this point on, the Leadwell V-40 iT and the
Haas UMC-750 machining centers are referred to as machine A and machine B,
respectively. Carbon SAE 1045 steel with a hardness of 190 HB (experimental
units) is utilized for the machining operation. The workpiece chemical
composition is presented in Table 1. A Fluke 1732 3-phase electrical energy
logger measures power consumption during the CNC machining processes. As
mentioned above, the cutting time has different stages (see Figure 1). In this
work, the energy consumption is obtained by integrating the electrical power
and considering the time of the effective cutting interval for each
experimental run. Additionally, the surface finish quality is evaluated using
a Tmteck TMR200 portable surface roughness checker, measuring Ra ($\mu$m). The
cutting tools are master cut from 1/2” coated tungsten carbide.

Table 1: Chemical composition of carbon steel SAE 1045 Element | C | Mo | P | S | Si
---|---|---|---|---|---
Wt% | 0.43-0.50 | 0.6-0.9 | 0.4 | 0.05 | 0.2-0.4

A factorial design has been selected, and the controllable factors with their
corresponding levels were chosen according to previous works [4, 28]. One of
the main purposes of this design is to obtain the optimal machining setting
conditions by considering the power consumption and roughness as response
variables. The experimental setup is summarized in Table 2.

Table 2: Experimental setup. |  |  | Levels
---|---|---|---
Factors | Units | Variable name | 1 | 2 | 3 | 4 | 5
Depth of cut | mm | $X_{1}$ | 1 | 1.5 | 2 | 2.5 | 3
Feed rate | mm/min | $X_{2}$ | 134 | 167.5 | 201 | 234.5 | 268
Spindle rate | RPM | $X_{3}$ | 950 | 1187.5 | 1425 | 1662.5 | 1900

Given the number of combinations from the levels of the three factors
presented in Table 2, 125 experiments are defined per machining center. Thus,
the total number of experimental runs is 250. Some pilot experiments are
carried out to set up the planning process, including the preprocessing and
postprocessing algorithm for the Sprutcam software used during the machining
process. Figure 2 shows the setup for side milling experimentation.

Figure 2: Experimental setup for the machining process

###  2.2 Statistical Analysis

The two outcomes obtained from the experiment are related to the same
experimental units; therefore, some significant level of correlation between
them could be present. Thus, to take into account this relationship in the
modelling, the seemingly unrelated regressions (SUR) model has been employed
[29, 30] to relate the outcomes to the covariates or fixed factors. The
specification of the model is as follows:

$\begin{split}log(\textbf{y}_{i})&=\beta_{i0}+I(m)\beta_{i7}+\sum_{j=1}^{3}(X_{j}\beta_{ij}+X_{j}^{2}\beta_{i,3+j}+X_{j}I(m)\beta_{i,j+7})+X_{1}X_{2}\beta_{11}+\\\
&\quad X_{1}X_{3}\beta_{12}+X_{2}X_{3}\beta_{13}+\epsilon_{i},\end{split}$ (1)

for $i=1,2$. Thus, $i=1$ and $i=2$, when the observed outcome is the roughness
($\textbf{y}_{1}$) and the power ($\textbf{y}_{2}$), respectively. $X_{1}$
(depth of cut), $X_{2}$ (feed rate), $X_{3}$ (spindle rate), represent the
controlled factors, and $I(m)$ is a dummy variable that defines a set of
equations for each machining center that has been evaluated. In this way, we
have

$I(m)=\begin{cases}1&\text{if it is the machine}\ \ m=B\\\ 0&\text{if it is
the machine}\ \ m=A,\end{cases}$

which consequently leads to

$log(\textbf{y}_{i})=\begin{cases}\beta_{i0}+\beta_{i7}+\sum_{j=1}^{3}(X_{j}(\beta_{ij}+\beta_{i,j+7})+X_{j}^{2}\beta_{i,3+j})+X_{1}X_{2}\beta_{11}+\\\
X_{1}X_{3}\beta_{12}+X_{2}X_{3}\beta_{13}+\epsilon_{i},&\text{for}\ \ m=B\\\
\beta_{i0}+\sum_{j=1}^{3}(X_{j}\beta_{ij}+X_{j}^{2}\beta_{i,3+j})+X_{1}X_{2}\beta_{11}+X_{1}X_{3}\beta_{12}+\\\
X_{2}X_{3}\beta_{13}+\epsilon_{i}&\text{for}\ \ m=A.\end{cases}$

The term $\epsilon_{i}$ is the random error related to all known and unknown
factors that have not been controlled in the experiment, and therefore not
considered in the deterministic component in the model (1). It is assumed that
$\epsilon_{i}\sim N(0,\sigma_{i}I_{n})$, which means that the random error is
normal distributed with mean zero and constant variance $\sigma_{i}$, where
$I_{n}$ is the $n\times n$ identity matrix. As we aim to model both equations
jointly, the vector $\bm{\epsilon}=(\epsilon_{1},\epsilon_{2})^{{}^{\prime}}$
is defined and it is assumed that $\bm{\epsilon}\sim N(0,\Sigma)$, where
$\Sigma$ is the covariance matrix that accounts for the individual variances
related to each outcome and the covariance structure between them. Given that
a Bayesian approach is being followed and there is no prior information
available for this experiment, non-informative or vague prior normal
distributions are set for the $\beta$ parameters and an inverse Wishart
distribution is set for the $\Sigma$ parameter. There are plenty of possible
justifications for the adoption of a Bayesian approach; as such, we highlight
the modelling flexibility it offers by considering the parameters as random
variables, the option to include prior information (prior to the experiment)
whenever available, and the advantage that its inferential properties hold for
any sample size, even small samples. For a better understanding of the
Bayesian paradigm, the interested reader is referred to [31] and [32]. To
assess the significance of the parameters in the model (1), high-density
intervals (HDI) are used, which are the Bayesian counterpart of the confidence
intervals (CI).

###  2.3 Optimal desing

Based on model (1), we aim to find the experimental conditions
$E=\\{X_{1},X_{2},X_{3}\\}$, which would produce an optimal outcome. What an
optimal response could mean depends on the interest of the experimenters. As
mentioned above, in this work, the interest is to find the experimental
conditions that minimize both the energy consumed (power) by the two machining
centers and the roughness value generated over the experimental units. To do
so, we define the minimum vector as $Y^{min}=(y_{1}^{min},y_{2}^{min})$, where
$y_{1}^{min}$ and $y_{2}^{min}$ are the minimum thresholds (for energy and
roughness, respectively) the experimenters are seeking to attain. Thus, within
this framework, the optimal design $E^{*}=\\{X_{1},X_{2},X_{3}\\}$ is defined
as follows:

$d(E^{*})=\min_{\forall E\in\mathcal{E}}d(\hat{Y},Y^{min}),$ (2)

where $\mathcal{E}$ is the set of all possible designs within the region where
the experiment is performed, $\hat{Y}$ is the predicted response obtained from
model (1), and $d(\cdot,\cdot)$ is a metric or distance function. To minimize
the function $d(\cdot,\cdot)$, we apply genetic optimization using derivatives
[33], specifically the version implemented in the R package rgenoud [34, 35].
gnoud is the package’s function that combines evolutionary algorithm methods
with a derivative-based (quasi-Newton) method to solve difficult optimization
problems. It is also possible to use this function for optimization problems
in which derivatives do not exist. It can also be applied to optimize
nonlinear or discontinuous functions in the variables of the function to be
optimized. One of the main advantages of using this method is that the genetic
optimizer rarely gets stuck at regions of local minimum points.

###  2.4 Bayesian ANOVA

ANOVA is a popular method widely used in data analysis related to experimental
designs. In this work, we employ the Bayesian ANOVA analysis proposed by [36].
It can be viewed as a graphical tool, that allows identification of the more
relevant factors in explaining or predicting the variability of an observed
response. This Bayesian version of the ANOVA is also based on a fixed/random-
effects model, which in this case is defined by:

$y_{ij}=\mu+\alpha_{j}^{D}+\alpha_{j}^{A}+\alpha_{j}^{V}+\epsilon_{ij},$ (3)

where $\mu$ is the global mean for the whole experiment, $\alpha_{j}^{f}$
represents the $j$-th treatment effect related to each factor controlled in
the experiment ($f=\\{D:\text{depth of cut},A:\text{feed rate},V:\text{spindle
rate}\\}$), for $j=1,\ldots,k_{f}$, where $k_{f}$ is the number of levels of
factor $f$. Thus, the finite population variance is given by

$S_{\bm{\alpha}^{f}}=\sqrt{\frac{1}{k_{f}-1}\bm{\alpha}^{{}^{\prime}f}\left[\textbf{I}_{k_{f}}-\frac{1}{k_{f}}\textbf{J}\right]\bm{\alpha}^{f}},$
(4)

where
$\bm{\alpha}^{f}=(\alpha_{1}^{f},\ldots,\alpha_{k_{f}}^{f})^{{}^{\prime}}$,
$\textbf{I}_{k_{f}}$ is the $k_{f}\times k_{f}$ identity matrix, and $\bm{J}$
is the $k_{f}\times k_{f}$ matrix of ones. As stated by [37], the traditional
mean square can be thought of simply as a point estimate of this quantity. In
contrast, the Bayesian approach provides a full posterior distribution for
$S_{\bm{\alpha}^{f}}$. Thus, the controlled factors with the higher
$S_{\bm{\alpha}^{f}}$ values will be the variables that are more relevant for
the explanation of the variability observed in both response variables. The
factors with lower or close to zero $S_{\bm{\alpha}^{f}}$ values can be
interpreted as irrelevant variables for the analysis. This analysis is used to
assess the relative importance of the (random or fixed) effects in a Bayesian
hierarchical linear model rather than testing their significance.

##  3 Results and discussion

In this section, the statistical analysis (introduced in section 2.2) of the
results related to the experiment described in section 2.1 are presented.
However, some descriptive analysis is first shown to give a better
understanding of the effect of the three controllable factors ($X_{1}$ (depth
of cut), $X_{2}$ (feed rate), $X_{3}$ (spindle rate)) over the two observed
responses (power consumption and roughness). Thus, in Figure 3 the patterns of
both power consumption and roughness are presented through the different
levels of spindle rate while fixing the higher and lower levels for depth of
cut ($X_{1}=1mm,X_{1}=3mm$) and feed rate ($X_{2}=134mm/min,X_{2}=268mm/min$).
For both machining centers a significant reduction of the power consumption
(up to 45% of electricity saving) when increasing the feed rate from
$134mm/min$ to $268mm/min$, for both levels of depth of cut, is clearly
observed from the dotted lines in Figure 3. On the other hand, from the same
figure, it can be seen that the roughness has a smooth variability as a
function of the spindle rate. In particular, when increasing the spindle rate
in machine A, the roughness value increases substantially for the lower level
of feed rate. This trend is corroborated in the surface plots presented in the
next section.

To perform a more general analysis of the influence of the three factors
$(X_{1},X_{2},X_{3})$ on energy consumption and surface finish (roughness),
surface plots were obtained from the fitting of model 1. Thus, Figures 4 and 5
show the response surfaces for machining centers A and B, respectively. These
graphs are obtained for pairwise factors while fixing the remaining factor on
its optimal level according to 2. When an increase in the feed rate is set,
the outcome for roughness also increases for machines A and B, whereas the
spindle rate does not show a significant effect on roughness for both machines
A and B. On the other hand, it is interesting to see the influence of the
depth of cut on the roughness, which is different for each machining center.
In the Leadwell V-40iT machining center (machine A), increasing the depth of
cut increases the roughness, reducing the quality of the surface finish (see
Figure 4 panels a, c). In contrast, in the Haas UMC-750 (machine B), when the
depth of cut increases, the roughness value decreases (see Figure 5 panels a
and c). That difference in the effect of the depth of cut on the observed
roughness is probably due to differences in vibration control for each
machine. However, the roughness responses are commonly known by their high
sensitivity to the operational conditions and the maintenance of the machining
center [15, 22, 21].

(a)

(b)

(c)

(d)

Figure 3: Power consumption and roughness for lower and higher experimental
levels at any spindle speed for both machining centers.

In terms of power consumption, the relationships among the three factors are
similar in both machining centers (see Figure 4 panels b, d, f and Figure 5
panels b, d, f). The feed rate which is the factor that determines the
duration

|
---|---
(a) | (b)
|
(c) | (d)
|
(e) | (f)
Figure 4: Variation of surface roughness and energy consumption for Leadwell V-40iT machining center (machine A). |
---|---
(a) | (b)
|
(c) | (d)
|
(e) | (f)
Figure 5: Variation of surface roughness and energy consumption for Haas
UMC-750 (machine B)

for each experiment, shows a direct relationship with the power consumption.
That is, when the feed rate increases, the power value increases. On the other
hand, the depth of cut shows a slight effect on the power values.

In Table 3, the significance analysis for the parameters related to the model
(1) is displayed. Their significance was analysed by using 95% high-density
intervals (HDIs), which are the Bayesian counterparts for 95% confidence
intervals (CIs). Thus, a parameter is classified as significant when its
interval excludes the zero value. The parameters related to $X_{1}$ (depth of
cut), $X_{2}$ (feed rate), $X_{3}$ (spindle rate), which represent the effects
of these factors over the whole observed power consumption and roughness
(i.e., without discriminating between machining centers), have a significant
effect on the observed power. For the observed roughness, the results are
similar except for $X_{3}$ (spindle rate), which has a non-significant effect
over the whole observed roughness (as observed in Figures 3, 4 and 5). The
coefficients related to the interactions $X_{1}I(m)$, $X_{2}I(m)$, and
$X_{2}I(m)$ represent the change of the overall effects related to $X_{1}$,
$X_{2}$, and $X_{3}$ when going from machining center $A$ to machining center
$B$. Those effects are not significant for the observed power, which may mean
that the power consumption mechanisms for the two machining centers are not
significantly different over the levels observed on $X_{1}$, $X_{2}$, and
$X_{3}$. On the other hand, those same effects for the observed roughness are
significant (at least for $X_{1}I(m)$ and $X_{2}I(m)$), which means that the
changes in roughness over the levels observed on $X_{1}$, $X_{2}$ are
statistically different between the two machining centers $A$ and $B$. For the
quadratic terms, only $X_{2}^{2}$ is shown to have a significant effect on
both responses. The practical interpretation for including nonlinear terms is
simply to capture patterns beyond marginal linearity. In the same vein, the
interaction terms $X_{1}X_{2}$, $X_{1}X_{3}$, and $X_{2}X_{3}$ are also
included, and among them, only the interaction $X_{1}X_{2}$ is shown to have a
significant effect on both responses. From this interaction term, one can
conclude that the particular effect of $X_{1}$ (at any level of depth of cut)
on both responses will depend on the level set for $X_{2}$ (feed rate) and
vice versa.

Table 3: Analysis of significance of the parameters from model (1) using high density intervals (HDI). | Response
---|---
| Roughness | Power consumption
Variable | HDI | Significant | HDI | Significant
$X_{1}$ | (0.054,0.224) | Yes | (0.031,0.059) | Yes
$X_{2}$ | (0.092,0.260) | Yes | (-0.229-0.201) | Yes
$X_{3}$ | (-0.112,0.058) | No | (0.002,0.031) | Yes
$X_{1}^{2}$ | (-0.012,0.131) | No | (-0.006,0.018) | No
$X_{2}^{2}$ | (-0.145,-0.001) | Yes | (0.010,0.034) | Yes
$X_{3}^{2}$ | (-0.054,0.088) | No | (-0.020,0.004) | No
$I(m)$ | (-0.486,-0.244) | Yes | (-0.571,-0.531) | Yes
$X_{1}I(m)$ | (-0.335,-0.094) | Yes | (-0.010,0.031) | No
$X_{2}I(m)$ | (-0.287,-0.046) | Yes | (-0.012,0.028) | No
$X_{3}I(m)$ | (-0.169,0.070) | No | (-0.016,0.025) | No
$X_{1}X_{2}$ | (-0.132,-0.011) | Yes | (0.002,0.022) | Yes
$X_{1}X_{3}$ | (-0.085,0.036) | No | (-0.007,0.013) | No
$X_{2}X_{3}$ | (-0.043,0.078) | No | (-0.005,0.015) | No

In Figure 6, the Bayesian ANOVA analysis for the three fixed factors
considered in this study are shown. As mentioned in Section 2.4, in contrast
to the classic ANOVA analysis, this is a graphical tool that allows
identifying the more relevant factors among those considered in the
experiment. From this analysis, one can observe that the feed rate is the most
relevant factor in explaining the observed variability in power consumed from
the two machining centers. In contrast, the most important factor in
explaining the observed variability in the roughness is the depth of cut. In
addition, for both responses, the spindle rate is the less relevant factor
among those analysed in this study in the levels described in Table 2.

|
---|---
(a) Bayesian Anova for power | (b) Bayesian Anova for roughness
Figure 6: Bayesian Anova analysis of the model 3. The inner and outer
intervals correspond to probability intervals of 50% and 95% respectively for
$f=\\{D:\text{depth of cut},A:\text{feed rate},V:\text{spindle rate}\\}$.

In Table 4 the results of the optimization process are shown, described in
section 2.3. The minimum threshold vectors for this process have been defined
as $Y^{min}=(40,0.33)$ and $Y^{min}=(24,0.28)$ for the machining centers
Leadwell V-40iT and Haas UMC-750, respectively. The experimenters define those
values, and in this case, they are taken from the minimum values that have
been observed in the experiment. The genetic algorithm has taken 20
generations to find the optimal conditions for both machining centers.

Table 4: Optimal operational conditions for the two machining centers found by the minimization of the functional $d(E^{*})$. | Optimal conditions
---|---
Machining center | Depth of cut | Feed rate | Spindle rate
Leadwell V-40iT | 1 | 268 | 950
Haas UMC-750 | 1 | 268 | 1111.512

In Table 5, Figures 7a and 8a present some important results related to the
predictive capability of the model presented in Equation (1). Thus, in Table 5
the predictions for roughness and power under the optimal operational
Conditions $E^{*}$ are shown. Outstanding performance is presented for the
specification defined in model (1) where low prediction errors are attained.
This same analysis has also been performed for the remaining 250 experimental
conditions observed in the experiment, and all of them yield prediction errors
below 5%.

Table 5: Observed and predicted responses under the optimal conditions reported in Table 4. | Machining center
---|---
Response | Leadwell V-40iT | Haas UMC-750
Power consumption |  |
Observed response | 40.28 | 24.86
Predicted response | 40.66 | 24.07
Error % | 0.01 | 0.03
Roughness |  |
Observed response | 1.18 | 1.35
Predicted response | 1.26 | 0.96
Error % | 0.07 | 0.28

In Figure 7a he boxplot analysis is presented for both observed roughness and
power consumption along with the location of the predicted and observed
responses under the optimal operational conditions presented in Table 4. From
those results, one can conclude that the optimization procedure has identified
the optimal design $E^{*}$ that at least has minimized one of the outcomes
(power). In addition, for the other response (roughness), it attained a
reasonable level of roughness. One has to bear in mind that the optimization
procedure described in Section 2.3 has been built to find the optimal design
that simultaneously minimizes both outcomes. Thus, the results presented in
Figure 5 ould mean that there are no design or operational conditions which
actually minimize both roughness and power consumption at the same time.
However, it is possible to find a design that minimizes one outcome (power
consumption) while reaching an acceptable result for the other (roughness).

(

(a)  Figure 7: Boxplot for each machining center’s observed power and
roughness (Machine A: Leadwell V-40iT, Machine B: Haas UMC-750).

a) Distribution for the observed power consumption

b) Distribution for the observed roughness

(

(b)

In Figure 8a the posterior predictive distributions are presented for both
roughness and power consumption under the optimal conditions reported in Table
4. These results can be interpreted as visual evidence of the predictive
performance of model (1), given that the observed or real responses lie inside
of the HDI. The interested reader can find the data and codes to reproduce

(

(a)

(

(b)

(

(e)  Figure 8: Densities of the posterior predictive distribution for each
machining center evaluated on the optimal design. The dotted lines represent
the observed responses reported in table 5 and the colored regions represent
the HDI of 95%.

a) b)
achining center A
c)  d)
achining center B

M

(c)

(

(d)

M

(f)

the main result from this study at github.com/JohnatanLAB.

##  4 Conclusions

This work presents an experimental study aimed at analysing the influence of
operational machining conditions on the performance of two machining centers.
Specifically, the influence of three controllable factors (depth of cut, feed
rate, and spindle rate) are studied over the power consumption and surface
finish (expressed through the superficial roughness) of two CNC vertical
machining centers (The Haas UMC-750 and Leadwell V-40iT). With a seemingly
unrelated regressions (SURs) model, it has been possible to simultaneously
model the two observed outcomes of the experiment in one single model, which
allows more degrees of freedom, i.e., more data available for the estimation
of each parameter. The resulting SUR model has been successfully employed to
find the optimal operational conditions by minimizing (2) using a genetic
algorithm. This optimization process suggests that the best performance values
for the feed rate ($268mm/min$) and depth of cut ($1mm$) factors are the same
for both machining centers. From the descriptive analysis and Bayesian ANOVA,
it is found that the feed rate is a dominant factor in controlling power
consumption. Similarly, the depth of cut reveals a significant relationship
with the roughness quality. This work has not considered the performance of
the cutting tools as one of the controllable factors. However, to obtain a
wide view from an energy efficiency perspective, the geometry and composition
of the cutting tools, including the tool path factor, should be considered
among the controllable factors in future research.

## acknowledgments

The authors gratefully acknowledge the financial support provided by
Institución Universitaria Pascual Bravo and Instituto Tecnológico
Metropolitano de Medellín.

## Conflicts of Interest

The authors declare no conflict of interest.

## Author Contributions

Conceptualization, J.C.J., J.S.R and M.I.A.; methodology, J.S.R, C.A.I.M.
E.J.N and J.C.J.; formal analysis, J.C.J., J.S.R., M.I.A. and C.A.I.M. ¸S.;
investigation, E.J.N, M.I.A and J.S.R.; resources, E.J.N., M.A.R and J.S.R.;
data curation, J.C.J., J.S.R. and M.I.A.; writing—original draft preparation,
J.C.J., J.S.R., and M.I.A.; writing—review and editing, C.A.I.M., J.S.R,
J.C.J., and M.A.R; administration, J.S.R. and M.I.A.; final revision, All
authors. All authors have read and agreed to the published version of the
manuscrip

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