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An interpretable predictive modelling framework for the
turning process by the use of a compensated fuzzy logic
system
Abdallah Alalawin
a
, Wafa’ H. AlAlaween
b
, Mohammad A. Shbool
b
,
Omar Abdallah
c
and Lina Al-Qatawneh
b
a
Department of Industrial Engineering, Faculty of Engineering, The Hashemite university, Zarqa, Jordan;
b
Department of Industrial Engineering, The University of Jordan, Amman, Jordan;
c
Dnata, Queen Alia
Airport, Amman, Jordan
ABSTRACT
This research presents a compensated fuzzy logic system that
integrates an interval type-2 fuzzy logic system (IT2FLS) with the
Gaussian mixture model (GMM) to model the turning process. First,
an IT2FLS is elicited to model the turning process by mapping its
input variables to the cutting force and the surface quality. Second,
the GMM is incorporated in the IT2FLS structure to compensate for
the error residuals. The idea of such an incorporation stems from
the fact that the majority of the models are constructed based on
the normality assumption of the error. The GMM is developed in a
way that renes the extracted rules and considers stochastic unmo-
delled behaviours. Validated on real experiments, it has been
demonstrated that the compensated fuzzy logic system has the
ability to accurately predict the cutting force and the surface qual-
ity; deal with uncertainties; and provide users with comprehensive
understanding of the turning process.
ARTICLE HISTORY
Received 27 July 2020
Accepted 5 April 2022
KEYWORDS
Compensated fuzzy logic
system; cutting force;
Gaussian mixture model;
interval type-2 fuzzy logic
system; surface quality;
turning process
1. Introduction
The manufacturing industry has witnessed a revolution upon a revolution. In the recent
Fourth Industrial Revolution (4IR), predictive analytics has been widely utilized to make
manufacturing systems predictable, flexible and also controllable (Lasi et al., 2014).
Among the various types of the manufacturing processes, the turning process, as
a machining one, can considerably benefit from the 4IR. In general, the turning process
is one of the four well-known metal cutting processes that are commonly utilized to
produce rotational and perhaps axisymmetric parts that usually include many features
(e.g. grooves, threads and holes; Lin et al., 2001). In such a process, a single-point cutting
tool or multi-point ones feed into a rotating workpiece to remove materials in the form of
chips to produce the parts required (Lin et al., 2001).
CONTACT Abdallah Alalawin abdallahh_ab@hu.edu.jo Department of Industrial Engineering, Faculty of
Engineering, The Hashemite university, Zarqa 13133, Jordan
PRODUCTION & MANUFACTURING RESEARCH
2022, VOL. 10, NO. 1, 89–107
https://doi.org/10.1080/21693277.2022.2064359
© 2022 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/
licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
Since the turning process is considered to be one of the key manufacturing processes,
a considerable research work has, therefore, been conducted to understand and model
such a process (Bhattacharya et al., 2009; Davim et al., 2008; Sharma et al., 2008). Some of
the related books and scientific research papers have focused on understanding the
influence of the various cutting variables (e.g. cutting (rotational) speed and feed rate)
and tool geometry (e.g. back rake angle) on the machined specimen’s surface roughness,
cutting force and tool wear (Bhattacharya et al., 2009; Davim et al., 2008; Sharma et al.,
2008). Such research papers and books have presented a comprehensive understanding
of the turning process and the parameters that affect it. However, there is a strong need to
enhance such an understanding by developing and implementing various predictive
modelling paradigms that have the ability to provide users with quantitative predictions
of the effect of the machined parameters on the output parameters (Sharma et al., 2008).
Therefore, modelling and predicting the cutting force of the turning process that is
essential for determining the power required by the cutting machine, and surface rough-
ness of the machined part have received a great deal of interest (S. Kumar et al., 2018).
The various predictive modelling approaches that have hitherto been implemented can
be either physical-driven models or data-based models (S. Kumar et al., 2018). Physical-
based models have been implemented to understand the process behaviour and to
develop the related mathematical equations that aim at finding the relationships between
the various cutting input parameters and the process responses (e.g. cutting force and
surface quality represented by surface roughness; Box & Draper, 1987). Due to the lack of
the related physical-based equations that can be utilized to represent the turning process
and with the huge advances in the recent computing power, data-driven paradigms have
found their way into many academic and industrial applications to represent input/
output relationships (S. Kumar et al., 2018). Multiple linear regression approaches have
been established to represent and predict various process parameters (e.g. cutting force,
tool wear, etc.) and the properties of a machined workpiece (e.g. surface roughness;
Bhardwaj et al., 2014; Uthayakumar et al., 2012). For instance, a regression model was
utilized to represent the relationship between the cutting force, as a dependent output,
and the feed rate, cutting speed and depth of cut, as independent inputs (Uthayakumar
et al., 2012). Such approaches cannot account for nonlinear relationships between inputs
and outputs, and cannot consider the sophisticated interactions that usually exist among
the process parameters (AlAlaween et al., 2016). Therefore, artificial neural networks
(ANNs) have been extensively utilized in modelling various machining processes (Liu &
Ko, 1990; Purushothaman & Srinivasa, 1994; Yao & Fang, 1993). For example, ANNs
were implemented to estimate the performance of the different metal cutting processes,
and to control the surface roughness characteristics (Grzesik & Brol, 2003;
Purushothaman & Srinivasa, 1994). In addition, a multilayer neural network was utilized
to predict the surface roughness of Ti-13Zr-13Nb alloy (Khanlou et al., 2016). In general,
these modelling paradigms are considered to be powerful interpolators that can be used
to represent complex input/output relationships in various processes, in particular, the
cutting processes (Grzesik & Brol, 2003). Because these models cannot provide users with
information about the process under investigation, they are commonly referred to as
black-box (Bishop, 2006), fuzzy logic systems (FLSs) have, therefore, been implemented
to map the process variables to the output parameters and the properties of a produced
part (Kamatala et al., 1996; Kuo & Cohen, 1998). For instance, the neuro-fuzzy paradigm
90 A. ALALAWIN ET AL.
was implemented to predict the surface roughness (Kamatala et al., 1996). It was
demonstrated that such a model outperformed other modelling approaches (e.g. regres-
sion models and ANNs) in terms of the predictive performance (Kamatala et al., 1996). In
addition to their ability to successfully represent complex relationships among the
examined inputs and outputs, FLSs, as data-driven models, can provide users with
a comprehensive understanding of the investigated process (AlAlaween et al., 2018).
Likewise, FLSs, as it is well-known, can effectively deal with uncertainties that result from
measurement uncertainties and from any uncontrollable variables, which may in some
cases have conflicting influences on the outputs under investigation. For instance, various
fuzzy logic based systems were developed to predict and optimize the cutting force and/
or the surface roughness for different materials while using different sets of the process
variables (Barzani et al., 2015; Marani et al., 2020, 2021). Likewise, the fuzzy logic system
was utilized to predict the material removal rate and surface roughness during diamond
surface grinding (Unune et al., 2018) However, the majority of the presented paradigms
(i.e. both the physical- and data-driven ones) are constructed based on the normality
assumption of the resulted errors (Yang et al., 2012). Such an assumption, which is, more
often than not, invalid in reality, can result in performance deterioration, this being due
to the unmodelled behaviour, stochastic or otherwise (Yang et al., 2012).
In this research paper, the main aim is to present an interpretable and more accurate
predictive modelling framework to successfully model the turning process. For this
purpose, a modelling structure called a compensated fuzzy logic system that integrates
deterministic and stochastic modelling paradigms is proposed. First, an interval type-2
fuzzy logic system (IT2FLS) is implemented to (i) represent the turning process; (ii)
predict the cutting force and surface roughness that is represented by the arithmetic
mean value (R
a
), it is worth emphasising at this stage that the cutting force plays an
important role in determining the fate of some of the outputs of the turning process in
terms of productivity and quality (i.e. it affects, for instance, the surface texture, which by
itself determines the fate of the machined parts, and tool wear; AlAlaween et al., 2021); (iii)
provide a simple understanding of such a process; and (iv) take into consideration the
uncertainties, which commonly surround the turning process. Then, the resulted error
residuals are characterised using the Gaussian mixture model (GMM), which compensates
for such resulted errors, to take into consideration any unmodelled behaviour, stochastic
or otherwise. The incorporation of the GMM in the modelling framework can improve the
modelling performance of the IT2FLS, this being due to its ability to (i) compensate for the
implicit normality assumption of the error residuals, and (ii) extract information that may
perhaps be hidden in the form of error residuals. The rest of the paper is organized as
follows: in Section 2, the set of experiments that were conducted using a lathe machine is
briefly mentioned. The background of the FLS, in particular, the IT2FLS and the obtained
results are summarized in Section 3. The GMM and its results are presented in Section 4.
Finally, Section 5 summarizes the conclusions of the whole paper.
2. Experimental work
In the turning process, there are many parameters that have an influence on the cutting
force and the quality of a machined specimen represented by the surface roughness
measured via the R
a
value. Since they are considered to be the most crucial parameters for
PRODUCTION & MANUFACTURING RESEARCH 91
the turning process (Bhattacharya et al., 2009; Davim et al., 2008; Sharma et al., 2008),
cutting (rotational) speed, depth of cut, feed rate and the use of lubricant were investi-
gated in this research work, as listed in Table 1. The levels of these parameters were also
defined in the same table. It is worth emphasizing at this stage that the levels of the four
variables investigated in this research work were defined by conducting a set of trial
experiments using the same material and cutting tool. Based on a full factorial design of
experiments, a total of 54 experiments were conducted using different input settings. It is
worth emphasising that each experiment was repeated three times. A lathe machine
(Colchester Master 2500, UK) was used to machine a cylindrical AISI D2 steel specimen
that has a diameter of 20 mm and length of 110 mm, as shown in Figure 1.
The cylindrical AISI D2 steel has a chemical composition of 1.5% Carbon, 0.3%
Silicon, 12% Chromium, 0.8% Molybdenum and 0.9% Vanadium, as summarized in
Table 1. Such a cylindrical specimen was placed with its longitudinal axis aligned with the
feed direction. The cutting tool that was utilized to conduct all experiments is Carbide
tipped insert coated with TiN. Once the cutting process was completed, the cutting force
and the surface roughness represented by R
a
were measured. The cutting force and the
surface roughness were measured to be used as target outputs to develop the proposed
model. In this research work, the cutting force was investigated because it is considered to
be one of the outputs that can be utilized to characterize the turning process in terms of
tool wear and more importantly surface texture (Piotrowska et al., 2009). The R
a
value
was measured using a Portable Stylus-Type Profilometer with LCD display (Senze
Instruments, Italy), while the cutting force was measured using two Dial Gauges Type
Table 1. The inputs and outputs of the turning process.
Inputs Inputs’ levels Outputs
Cutting speed 175, 235 and 320 (rpm) Cutting force (KGF)
Depth of cut 0.1, 0.15 and 0.2 (mm) Surface roughness* (µm)
Feed rate 0.05, 0.1 and 0.16 (mm/rpm)
Use of lubricant Dry and Cutting fluid (Zinol, UAE)
Chemical composition 1.5% Carbon, 0.3% Silicon, 12% Chromium, 0.8% Molybdenum and 0.9% Vanadium
*Surface roughness was measured via the arithmetic mean value (R
a
).
Figure 1. The lathe machine (Colchester Master 2500, UK).
92 A. ALALAWIN ET AL.
60/0.002 mm (TecQuipment, UK). It is worth mentioning that the two Dial Gauges were
utilized to measure the tool/tool holder deflection that was proportional to the magnitude
of the cutting force as explained in (H. Kumar & Kumar, 2011). The average values of the
cutting force and surface roughness for the three replicates of each experiment were
determined in this research work. The variability values for the cutting force and the
surface roughness were in the range of 1.7 to 4.5 and in the range of 0.6 to 5.6,
respectively.
In order to evaluate the strength of the linear relationships between the investigated
quantitative inputs and outputs, the statistical linear correlation analysis was conducted
between the examined process variables and the measured cutting force and the surface
roughness represented by R
a
. Reasonable correlation coefficient values among most of
the investigated parameters can be noticed in Table 2. However, some of the examined
variables have different correlation coefficient values between the experiments that were
carried out without the use of lubricant/cutting fluid (i.e. dry process) and the ones that
were conducted with the use of lubricant. For instance, the relationship between the
cutting speed and the cutting force without the use of cutting fluid is relatively stronger
when compared to the same relationship when the cutting fluid was used. By implement-
ing the analysis of variance test, as a statistical test, it was noticeable that the use of
lubricant, as a classical input parameter, has a significant influence on both the cutting
force and the surface roughness represented by the R
a
values (i.e. the P-values are less
than 0.05).
3. Interval type-2 fuzzy logic systems
3.1. Model development
With the huge advances in recent computing power, the development of computational
intelligence has positive effects on several areas including, but not limited to, healthcare
and manufacturing. Utilizing computer systems has also changed how researchers think
in industry and academia. The observed/collected data are, therefore, used to develop and
construct data-driven modelling approaches that mimic the human way of thinking.
Such modelling paradigms can complement or replace the so-called physical-based
models, in particular, for those processes where such models do not exist or they can
be too complex to derive. Therefore, a plethora of data-driven paradigms (e.g. regression
models and ANNs) have hitherto been developed and implemented in various research
areas, such as healthcare, manufacturing and marine technology (AlAlaween et al., 2017;
Shahani et al., 2009). Despite their powerful algorithms, some of the presented models
(e.g. regression paradigms) are incapable of representing complex highly nonlinear
Table 2. The correlation coefficients.
Outputs Dry Cutting fluid
Inputs Cutting Force Surface roughness* Cutting Force Surface roughness*
Cutting speed −0.34 −0.38 0.16 −0.02
Depth of cut 0.16 −0.17 0.42 0.13
Feed rate 0.31 0.32 0.22 0.46
*Surface roughness was measured via the arithmetic mean value (R
a
).
PRODUCTION & MANUFACTURING RESEARCH 93
relationships. Likewise, some of these models, in particular ANNs, are referred to as
black-box ones, this being due to the low interpretability of such models(AlAlaween
et al., 2018) . Therefore, the fuzzy logic system (FLS) has hitherto been utilized in various
applications to develop an interpretable model that can efficiently take into consideration
uncertainties (i.e. measurement uncertainties and uncertainties that may result from any
uncontrollable and difficult to consider variables; AlAlaween et al., 2018).
In general, FLSs can usually be described by fuzzy sets. In general, type-1 and type-2
are the common types of these fuzzy sets. The system that its antecedents and consequent
of the rules are characterised by the former fuzzy sets is referred to as the type-1 fuzzy
logic system (T1FLS). Whereas a type-2 fuzzy logic system (T2FLS) is the one that at least
one of its antecedents and consequent of the rules is characterised by the latter fuzzy sets
whose membership functions are fuzzy (Karnik & Mendel, 2001). The T2FLS, as it is
known, is able to handle uncertainties more effectively and efficiently compared to the
T1FLS. However, as it is well known, such a system is considered to be computationally
expensive. Thus, an interval type-2 fuzzy logic system (IT2FLS) was presented and
thenceforth it has been implemented and utilized (Tan & Chua, 2007). The correspond-
ing fuzzy set for the IT2FLS can be written as presented in Equation (1; Tan & Chua,
2007):
A
%¼ðx2X
1muðu2Jx½0;1
1=x;uð Þ (1)
where x and X represent the primary variable and its measurement domain, respectively.
The parameters J
x
and u stand for the primary membership degree and the secondary
variable; where u belongs to J
x
at each x belongs to X, respectively.
The IT2FLS framework is shown in Figure 2. As shown in such a figure, the first step is
to fuzzify the crisp inputs (x
1
, x
2
. . . x
n
) into type-2 fuzzy sets (~
Ai
j); where ~
Ai
j represents the
ith fuzzy set for the jth parameter. In the fuzzification step, the membership functions
both the upper and lower μ~
Ai
j;
μ~
Ai
j
h iare determined. The smoothness and continuity of
the Gaussian function commonly allow the fuzzy system to be implemented in the form
Figure 2. The framework of IT2FLS.
94 A. ALALAWIN ET AL.
of a universal approximator. Therefore, such a function is a common choice for the
membership function. The Gaussian membership function can simply be written as
represented in Equation (2) (Tan & Chua, 2007).
μi
jðxjÞ ¼ exp 1
2
xjmi
j
σi
j
!" #;mi
j2mi
j1;mi
j2
h i (2)
where mi
j and σi
j represent the mean and the standard deviation values of the ith set for
the jth variable, respectively. The subscript number is used to distinguish the lower mean
(mi
j1) from the upper one (mi
j2). The area shaded between the lower and the upper
membership functions is usually known as the footprint of uncertainty.
The input fuzzy sets are then mapped to the output fuzzy sets by combining the
defined rules, such a process is called the inference process. Commonly, the rules can be
extracted from an available data set (i.e. the experimental data) or can be noticed and
provided by experts. These rules are usually expressed in the form of a set of IF-THEN
rules, as follows:
Rule
i
: IF x
1
is ~
Ai
1 . . . and x
n
is ~
Ai
n, THEN y is ~
Bi.
where Ai
j and Bi stand for the membership functions of jth antecedent of the ith rule
and the consequent of the same rule, respectively. It is worth emphasising that the rules
for the T1FLS and the T2FLS have a similar form, the only difference is related to the
membership functions nature. In the presented research work, the Mamdani fuzzy
system is utilized. In such a system, B
i
is represented by a membership function.
Membership functions that are generally expressed by words (e.g. low and medium)
are usually described by fuzzy sets, which commonly represent subjective information
provided about an examined process. The output of the inference process is the type-2
output fuzzy sets that are then reduced into type-1 ones. In such a step, the lower and
upper limits are usually determined using Karnik-Mendel (KM) algorithm (Karnik &
Mendel, 2001). It is worth emphasizing that the type reduction step incurs most of the
computational effort needed. Finally, a defuzzification process is utilized to calculate
a crisp output by simply calculating the average value (Tan & Chua, 2007).
3.2. Results and discussion
In this research work, the IT2FLS was implemented to predict the cutting force and the
quality of a machined specimen represented by the surface roughness measured via the
R
a
value. The IT2FLS was employed in this research work because it can, in general, (i)
represent highly nonlinear input/output relationships; (ii) handle the uncertainties that
may surround the process effectively; and (iii) provide users with a simple and linguistic
understanding of the process examined in the form of If/Then rules. In order to develop
such a model and due to the limited amount of data points, the data collected were
classified into two sets only: training set that contains 38 experiments and testing set that
includes 16 experiments. The training data set is usually utilized to extract rules and, thus,
it allows the paradigm to learn the input/output relationships, whereas the testing one is
utilized to examine the FLS generalization capabilities. It is worth mentioning at this
stage that different division methods have hitherto been investigated in the related
literature (e.g. the 10-fold cross-validation technique and random division method). In
PRODUCTION & MANUFACTURING RESEARCH 95
this research paper, it was found that dividing the data randomly into training and testing
sets was the best and simple method. To successfully model the turning process, the
nature of the examined input parameters should be understood (i.e. discrete or contin-
uous). In this research work, all the input parameters were considered as continuous ones
except the use of lubricant parameter, which was considered as a crisp one. For a specific
number of rules, and by using the Gaussian membership function, the IT2FLS para-
meters, such as mean and standard deviation, were initialized by utilizing the interval
type-2 fuzzy clustering algorithm (Rubio & Castillo, 2013). Then, they were optimized by
implementing the steepest descent algorithm that is commonly integrated with the back-
propagation network (Karnik & Mendel, 2001). The best number of rules was the one
that resulted in the minimum difference between the target and predicted values. Such
a difference was estimated by the root mean square error (RMSE).
The performance of the IT2FLS for the cutting force is presented in Figure 3, with
a RMSE (training, testing) = [1.197, 1.230]. Obviously, one can notice that the RMSE
value for the testing data set is slightly greater than the one for the training data set
(approximately 3% higher). This can indicate that an overtraining problem has occurred.
Figure 3. The IT2FLS for the cutting force (KGF): (a) Training, (b) Testing (with a 90% confidence
interval).
96 A. ALALAWIN ET AL.
However, it does not seem to be the case in this model, where this difference is associated
to the cutting force values in both the training and the testing data sets. To elucidate
further, three data points out of 16 in the testing data set have values that are greater than
20KGF, thus, the error values are quite large, but less than 10% of the target value, these
points can significantly affect the RMSE value in the testing set. This can be demonstrated
by estimating the coefficient of determination, R
2
(training, testing) = [0.888, 0.883].
Furthermore, it is noticeable that the majority of the predicted values fit properly in
a 90% confidence interval, as shown in Figure 3.
Out of a total of 6, two rules, as examples, are represented in Figure 4, where the
footprint of uncertainty is represented by the shaded area, and the corresponding
linguistic forms of such rules can be as follows:
Rule 1: IF no cutting uid is used and cutting speed is medium and feed rate is small and
depth of cut is small, THEN the cutting force is medium.
Rule 2: IF cutting uid is used and cutting speed is high and feed rate is high and depth of
cut is medium, THEN the cutting force is high.
Examples of the response surfaces for the cutting force using two parameters at a time
are shown in Figure 5. It is noticeable that the cutting force is a non-linear function of the
cutting speed, depth of cut and feed rate. It is also noticeable that at a low level of depth of
cut (less than 0.15 mm) and high cutting speed the cutting force is low, whereas the
increase of the depth of cut increases the cutting force. Furthermore, when the depth of
cut is in the range of 0.15 mm to 0.2 mm, the cutting force is reaching a saturation level
when a low level of feed rate (less than 0.08 mm/rpm) is used. In addition, a high level of
cutting force can be noticed when both the levels of feed rate and depth of cut are high.
In a similar way, the IT2FLS was also implemented to predict the quality of
a machined part that is represented by the surface roughness measured via the R
a
value. By using eight rules, the IT2FLS performance for the R
a
values is presented in
Figure 4. The rule base of IT2FLS for the cutting force.
PRODUCTION & MANUFACTURING RESEARCH 97
Figure 6, with RMSE (training, testing) = [1.223, 1.241] and R
2
(training, testing) = [0.856,
0.852]. One can notice that the performance measures for the surface roughness are low
when compared to the performance measures for the cutting force, this being due to the
high uncertainties in measuring the surface roughness.
Figure 5. Examples of the response surfaces for the cutting force.
Figure 6. The IT2FLS for the R
a
value (μm): (a) Training, (b) Testing (with a 90% confidence interval).
98 A. ALALAWIN ET AL.
The IT2FLS requires more computational efforts (i.e. computationally expensive)
when compared to T1FLS, such a fact raises the question of whether such computational
efforts have resulted in a superior paradigm for the turning process. Thus, the T1FLS was
utilized to predict the cutting force and the surface roughness represented by the R
a
of the
machined specimen. The performance measures represented by the R
2
and the RMSE
values are summarized in Table 3. In such a table, one can notice that for both the cutting
force and the surface roughness, the predictive modelling performance of the IT2FLS
presented also in Table 3 is superior to that of the T1FLS; this being due to the fact that
the IT2FLS can systematically deal with uncertainties more effectively compared to
T1FLS. It is also worth noting that the predictive modelling performance for the surface
roughness measured via the R
a
value is worse than the one for the cutting force for both
T1FLS and IT2FLS.
4. Gaussian mixture model
4.1. Model development
Most of the modelling paradigms, including a T1FLS and a T2FLS, are based on the
assumption that the set of error residuals is distributed normally (Yang et al., 2012). On
real-world applications with noisy measurable or non-measurable factors, this assump-
tion, in fact, may not always be valid and, as a result, may lead to a loss of valuable
information and to a paradigm with sub-optimal parameters (AlAlaween et al., 2018).
Therefore, various modelling algorithms have hitherto been presented and implemented
to extract such valuable information and improve the modelling results by characterizing
the error residuals (Yang et al., 2012). For instance, the stochastic Gaussian mixture
model (GMM) that can usually be presented as a linear combination of a number of the
Gaussian components, has been implemented to refine the predictive performance of
a model by providing a deeper interpretation of the density function. In the fuzzy logic
systems, all of the presented algorithms including the GMM algorithm will, however,
change the predicted output values without changing the extracted rules (i.e. the
Table 3. The performances of the models represented by RMSE and R
2
.
Output
Cutting force
(KGF)
Surface roughness*
(µm)
Models Train Test Train Test
IT2FLS R
2
0.888 0.883 0.856 0.852
RMSE 1.197 1.23 1.223 1.241
IT2FLS with bias compensation R
2
0.949 0.96 0.924 0.92
RMSE 0.805 0.803 1.108 1.112
T1FLS R
2
0.815 0.809 0.765 0.771
RMSE 1.289 1.283 1.374 1.38
T1FLS with bias compensation R
2
0.843 0.842 0.827 0.82
RMSE 1.201 1.198 1.311 1.309
Fuzzy logic based on sub-clustering R
2
0.802 0.812 0.755 0.761
RMSE 1.329 1.192 1.381 1.418
Taguchi Artificial Neural Network Hybrid with Genetic algorithm R
2
0.782 0.779 0.721 0.713
RMSE 1.632 1.592 1.824 1.762
*Surface roughness was measured via the arithmetic mean value (R
a
).
PRODUCTION & MANUFACTURING RESEARCH 99
consequents of the extracted rules). Consequently, these rules can no longer represent the
process under investigation (AlAlaween et al., 2018) . In this research work, the GMM is,
therefore, incorporated in the fuzzy logic system, by such an incorporation the informa-
tive extracted rules are refined. The GMM was implemented because of its ability to
represent the probability density function with a rational accuracy using the best number
of Gaussian elements. The incorporation of the IT2FLS and the GMM is schematically
represented in Figure 7.
As shown in Figure 7, the first step in such an incorporation is to select the set of the
input variables that can be utilized to characterize the error. The optimal parameters of
the GMM, namely; mean, covariance and mixing coefficient for each Gaussian element,
are then determined by optimizing the well-known log-likelihood function. The set of
optimal parameters can be written as shown in Equation (3; Bishop, 2006):
ϕðzijÞ ¼ πjNxe
ijμe
j;e
j
P
J
j¼1
πiNxe
ijμe
j;e
j
;"j
μe
j¼P
I
i¼1
ϕðzijÞxe
i
P
I
i¼1
ϕðzijÞ
e
j¼P
I
i¼1
ϕðzijÞxe
iμe
j
xe
iμe
j
T
P
I
i¼1
ϕðzijÞ
πj¼P
I
i¼1
ϕðzijÞ
I
8
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>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
;"j
(3)
where xe
i is the ith data vector that need to be included in the errors characterization
GMM. The parameters μe
j, πj and e
j stand for the mean, the mixing coefficient and the
covariance of the jth Gaussian element, respectively. The parameter ϕðzijÞrepresents
Figure 7. The incorporation of the fuzzy logic system and the modified GMM algorithm.
100 A. ALALAWIN ET AL.
the probability of the ith data vector belongs to the jth Gaussian element, thus, zij, as
a J-dimensional binary parameter, is assigned a value of one when the ith vector
belongs to the jth component where the other elements are assigned zero values.
Since defining the set of optimal parameters analytically is not an easy task, the
Expectation Maximization (EM) algorithm is usually implemented (Bishop, 2006).
First, the parameters μe
j, e
j, πi are carefully initialized using a clustering algorithm.
In this research work, the K-means algorithm was utilized to initialized such para-
meters. Such a step is followed by determining the value of ϕðzijÞ, such a step is called
the E-step. Second, in the M-step, the estimated value of ϕðzijÞis used to update the
parameters, which are used to re-estimate the ϕðzij Þvalue. The described procedure is
continued until the EM algorithm converges or the pre-defined number of iterations is
reached (Bishop, 2006). Commonly, the optimal number of the Gaussian elements is
unknown, therefore, in this research paper, the performance measure that is employed
to select such a number is the Bayesian information criterion (BIC; Rogers & Girolami,
2016).
The conditional error mean and the standard deviation can be estimated using
numerical methods (Bishop, 2006). The optimal conditional mean value, as an indication
of the exist bias, is then added up to the consequent mean of the related rule, by this the
GMM algorithm can compensate for the bias. Finally, this step is, then, followed by the
defuzzification process to determine the crisp output. Such steps are summarized in the
flowchart presented in Figure 8.
4.2. Results and discussion
The main effects of the investigated process variables were considered by developing and
optimizing the IT2FLS. Thus, only two variables out of a total of four were utilized to
implement the GMM algorithm to compensate for the bias that could result from the
error normality assumption. Different combinations of input variables were tested, the
combination that was finally chosen was the one that resulted in the maximum error
compensation (i.e. the minimum RMSE value). For the cutting force, the feed rate and
the depth of cut were employed, in addition to the vector that consists of the errors
resulted from the IT2FLS, to construct the GMM. The training data set was utilized to
train such a model, whereas the testing one was kept hidden during the training process.
By using six Gaussian elements, the predictive performance for the cutting force for the
training and testing data sets is shown in Figure 9. The corresponding R
2
[training,
testing] value is [0.949, 0.960].
An overall improvement of approximately 7% in R
2
value proves the ability of the
GMM to detect possible unmodelled stochastic or deterministic behaviour. Figure 10
shows the two rules presented in Section 3 above after bias compensation (i.e. incorpor-
ating the GMM in the IT2FLS structure). It is noticeable that the antecedents of the two
rules presented in Figure 10 and the ones presented in Figure 4 are the same, the
difference is only in the consequents. To illustrate, the consequents of the first
and second rules were amended by approximately 0.9KFG to the right and 0.85KFG to
the left, respectively. It is worth emphasising that such changes in the consequent mean
values had no effect on the linguistic forms of these rules.
PRODUCTION & MANUFACTURING RESEARCH 101
Similarly, the GMM algorithm presented in this work to refine the FLS rules was also
implemented to refine the IT2FLS that was developed to model the surface roughness,
which is represented by the R
a
value of the produced product. Using seven Gaussian
components, the performance measure values for the surface roughness are listed in
Table 3. It is obvious that an improvement value of approximately 8% in the R
2
value was
gained. It is also evident that the predictive performance of the compensated IT2FLS that
was developed for the cutting force is better than the one of the compensated IT2FLS that
was constructed for the surface roughness represented by the R
a
value, whereas the
improvement value for the surface roughness is slightly large when compared to the
improvement value for the cutting force.
Figure 8. Flowchart of the implementation of the GMM algorithm.
102 A. ALALAWIN ET AL.
For comparison purposes, the GMM algorithm was utilized to amend the rules that were
extracted by the T1FLSs that were developed for both the cutting force and the surface
roughness, leading to significant improvement values. The performance measures repre-
sented by RMSE and R
2
are listed in Table 3. However, it is worth noting at this stage that the
model integrating the IT2FLS with the GMM algorithm outperforms the model integrating
the T1FLS with the GMM algorithm, with significant improvement values of approximately
13% and 12% in R
2
for the cutting force and the surface roughness, respectively. Such
expected results indicate that the IT2FLS can deal with uncertainties better than the T1FLS
and can, consequently, lead to a better predictive performance. Likewise, the traditional
GMM presented in (Yang et al., 2012), which is usually utilized to refine the data points
instead of rules similar to the other error characterization paradigms, was used to refine the
IT2FLS, it was demonstrated that the algorithm presented in this research paper, where the
extracted rules are refined, is superior to the traditional one, with improvement values of
approximately 4% and 5% in the R
2
value for the cutting force and the surface roughness,
respectively. In addition, the transparency of the IT2FLS was kept and maintained during the
proposed error characterization approach. In addition, two proposed algorithms, namely,
Figure 9. The prediction performance for the cutting force after bias compensation (with a 90%
confidence interval).
PRODUCTION & MANUFACTURING RESEARCH 103
a fuzzy logic based on sub-clustering approach (Rodić et al., 2021) and Taguchi artificial
neural network hybrid with genetic algorithm (Nukman et al., 2013) were employed in this
research work for comparison purposes. The results obtained are summarized in Table 3. It
is noticeable that the results of the Fuzzy logic based on sub-clustering approach are close to
the ones of the T1FLS. Furthermore, the results of the Taguchi artificial neural network
hybrid with genetic algorithm are not as expected, this being due to the fact that the artificial
neural network, in general, cannot deal with uncertainties.
In summary, the presented modelling framework, which integrates the IT2FLS
and the GMM algorithm, has superior predictive performance when compared to
the well-known IT2FLS and T1FLS as presented in Table 3. In addition, the
proposed model and the well-known IT2FLS and T1FLS are considered to be
transparent models that can provide users with a simple understanding of the
process under examination and that can deal with uncertainties intrinsically.
However, the proposed modelling framework requires more computational efforts
(i.e. computationally expensive) when compared to IT2FLS and T1FLS. Such
computational efforts have resulted in a superior predictive performance for the
cutting force and the surface roughness. After being successfully trained, the
presented model, as a data driven model, can also be used to predict the cutting
force and the surface roughness for new materials and perhaps new variables. In
addition, the proposed framework represents a promising development not only in
the manufacturing industry but also in other industries, where one needs to
develop a predictive model that can (i) accurately predict the properties of
a produced product in a way that can guarantee and perhaps optimize the quality
attributes that can be critical, (ii) provide and maintain a simple and comprehen-
sive linguistic understating of the process under investigation, and (iii) character-
ize the error residuals to compensate for the normality assumption (i.e. the error is
normally distributed) and, consequently, model the stochastic and deterministic
behaviours.
Figure 10. The rule base of IT2FLS for the cutting force after bias compensation.
104 A. ALALAWIN ET AL.
5. Conclusions
The main aim of the presented research work was to develop an interpretable and
more accurate predictive structure for the turning process. The developed structure
incorporated a Gaussian mixture model (GMM) in the structure of an interval type-2
fuzzy logic system (IT2FLS). The IT2FLS was implemented first to represent the
turning process by mapping the process parameters to the cutting force and the
surface quality of a machined specimen. The IT2FLS was able to successfully predict
the cutting force and the surface roughness and to deal with any uncertainties. In
addition, it provided informative rules that can be easily understood and utilized to
control the turning process. Since, more accurate predictions of the examined attri-
butes are, more often than not, desired, the GMM was then utilized to characterise the
resulted error residuals by refining the extracted informative rules to compensate for
any potential biases and to, consequently, improve the predictive performance. An
overall improvement of 7% was gained by incorporating the GMM in the IT2FLS
structure.
Disclosure statement
No potential conflict of interest was reported by the author(s).
ORCID
Wafa’ H. AlAlaween http://orcid.org/0000-0001-8661-3606
Mohammad A. Shbool http://orcid.org/0000-0002-9413-7985
Lina Al-Qatawneh http://orcid.org/0000-0001-9314-7914
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