Citation: Rezgui, M.A.; Trabelsi, A.;
Barbana, N.; Ben Youssef, A.;
Al-Addous, M. Static Robust Design
Optimization Using the Stochastic
Frontier Method: A Case Study of
Pulsed EPD Process on TiO2Films.
Inventions 2024,9, 31. https://
doi.org/10.3390/inventions9020031
Academic Editor: Kambiz Vafai
Received: 28 January 2024
Revised: 1 March 2024
Accepted: 4 March 2024
Published: 8 March 2024
Copyright: © 2024 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
inventions
Article
Static Robust Design Optimization Using the Stochastic Frontier
Method: A Case Study of Pulsed EPD Process on TiO2Films
Mohamed Ali Rezgui 1, Ali Trabelsi 1, Nesrine Barbana 2,* , Adel Ben Youssef 3and Mohammad Al-Addous 4,5
1
Laboratory of Mechanics, Production and Energy, Higher National Engineering School of Tunis, University of
2Department of Environmental Process Engineering, Institute of Environmental Technology, Technische
Universität Berlin, 10623 Berlin, Germany
3Higher National Engineering School of Tunis, University of Tunis, Tunis 1008, Tunisia;
4Department of Energy Engineering, School of Natural Resources Engineering and Management, German
5Fraunhofer-Institut für Solare Energiesysteme, ISE, 79110 Freiburg, Germany
*Correspondence: [email protected]
Abstract: This paper aims to optimize a pulsed electrophoretic deposition (EPD) process for TiO
2
films. This is accomplished by determining the optimal configuration of the coating parameters from
a robust optimization perspective. The experimental study uses a composite central design (CCD)
with four control factors, i.e., the initial concentration (x
1
in g/L), the deposition time (x
2
in s), the
duty cycle (x
3
in %), and the voltage (x
4
in V). The process responses that should all be maximized
are the photocatalytic efficiency of the thin film (De) and three critical charges, which characterize the
adhesion failure, i.e., L
C1
: the load at which the first cracks occurred; L
C2
: the load at which the film
starts to delaminate at the edge level of the scratch track; and L
C3
: the load when the damage of the
film exceeds 50%. This paper compares the robust optimization design of the EPD process using two
methods: the robust design of processes and products using the stochastic frontier (RDPP-SF) and the
surface response and desirability function methods. The findings show that the RDPP-SF method is
superior to the response surface–desirability method for the process responses De and L
C2
because of
non-natural sources of variation; however, both methods perform comparably well while analyzing
the L
C1
and L
C3
responses, which are subjected to pure random variability. The parameters setting
for the process robust optimization are met in run 25 (x
1
= 14 g/L, x
2
= 150 s, x
3
= 50%, and x
4
40 V).
Keywords: pulsed electrophoretic deposition; titanium dioxide film; composite central design;
Taguchi method; multi-response problems; static robust design; stochastic frontier; hypothesis tests
1. Introduction
Various coating techniques have been employed to deposit titanium dioxides TiO
2
,
such as spin coating [
1
], plasma-enhanced chemical vapor deposition (PECVD) [
2
], anodiza-
tion [
3
], and electrophoretic deposition (EPD) [
4
,
5
]. However, electrophoretic deposition
(EPD) is more popular for depositing TiO
2
coatings due to its low cost, basic equipment re-
quirements, short processing time, high productivity, and simple setup. The EPD technique
utilizes electrostatically stabilized suspensions, which are often organic [
6
] or mixtures
of organic solvents and water [
7
]. In an aqueous medium, the electrochemical reactions
at the surface of the electrodes can cause the formation of bubbles in the deposit, which
may make adhesion difficult. One way to mitigate this problem is to create a conver-
sion layer (CL) before the coating using conventional techniques for chemical etching of
the surface [
4
]. Stainless steel (SS-316L) is employed as a substrate for titanium dioxide
(TiO
2
) films because of its electrical and mechanical properties as well as its low corrosion
rate [
8
–
10
]. However, the adhesion between the TiO
2
and bare SS-316L is poor due to their
Inventions 2024,9, 31. https://doi.org/10.3390/inventions9020031 https://www.mdpi.com/journal/inventions
Inventions 2024,9, 31 2 of 14
chemical incompatibility [
11
,
12
]. A few research studies have tackled the effect of the EPD
operational factors on the decolorization efficiency and adhesion of the film to find the
optimal operational conditions for the deposition of the TiO
2
layer. Studies have shown
that the CL has led to the improvement of film adhesion and resulted in a degradation
percentage of 60% [4].
This paper extends previous research on optimizing the EPD process for the deposition
of coatings by investigating the factor setting in the context of a robust optimization
perspective. This study employs the static version of the robust design of products and
processes using the stochastic frontier method (RDPP-SF), as proposed by Trabelsi and
Rezgui [
13
] and Rezgui and Trabelsi [
14
], to estimate both the random part-to-part and
systematic variations, which occur in the pulsed EPD process. Using this approach, this
study aims to find the optimal setting of the process parameters, which yields a sustainable
improvement in the coating quality by reducing the global variability in the EPD process.
1.1. Robust Design of Products and Processes
Robust design optimization, referred to as design for Six Sigma (DFSS), seeks to establish
process parameters at specific levels such that the process output is close to the optimal
while noise factors are acting. The Taguchi method [
15
] for robust optimization of products
and processes is a popular approach, which is based on the signal-to-noise (S/N) ratio
metric. The engineering and research community embraces the method because of its easier
implementation and engineering common sense. It operates in two major steps: (i) find out
the best set of the control parameters, which yields tight response variation; and then (ii) bring
the average closer to the target value while operating under the environmental noise factors.
Nonetheless, the Taguchi method is frequently criticized, primarily for its procedural scheme
and signal-to-noise ratio. As a result, alternative methods have been developed. The RDPP-SF
method [13,14] is just one contribution in this regard.
Del Castillo et al. [
16
] studied multi-response functions with non-differentiable points.
They presented modified desirability functions, which allow any gradient-based optimization
method to maximize overall desirability. Chen [
17
] devised a method where the S/N ratio
is transformed into satisfactory indices, which are fitted to the multi-regression model in
multi-response problems. Su and Tong [
18
] developed a method that is based on the principal
component analysis (PCA); thereby, the original responses are collapsed into a few uncorre-
lated responses to optimize multi-response problems using the Taguchi method. Tong and
Su [
19
] used a fuzzy logic technique to cope with constraints and multi-criteria; hence, the
optimization problem is reverted to a multiple-attribute decision-making problem.
Antony [
20
] used Taguchi’s quadratic loss function to optimize multi-response problems
in manufacturing. Kim and Lin [
21
] used a modified exponential desirability function to
optimize multi-response problems. Liao and Chen [
22
] used data envelopment analysis (DEA)
ranking to optimize multi-response problems. Moreover, Liao [
23
] developed a procedural
scheme, which is based on the process capability ratio (PCR) theory and the theory of order
preference by similarity to the ideal solution (TOPSIS), to optimize multi-response problems.
Hsu et al. [
24
] utilized the artificial neural network (ANN) technique and exponential desir-
ability function to optimize the performance of the broadband tap coupler. Ortiz et al. [
25
]
developed a multiple-response solution technique using a genetic algorithm, which was
combined with an unconstrained desirability function. Liao [
26
] used the ANN methodology
and data envelopment analysis (DEA) to perform an optimization of multi-response problems.
Fung and Kang [
27
] optimized the injection molding process for the friction properties of
polybutylene terephthalate using the Taguchi method and PCA technique. Liao [
28
] devel-
oped an original method based on the weighted principal component technique to address
the optimization of multi-response problems. Liao et al. [
29
] transformed multi-response
optimization into performance indices using the canonical correlation technique; therefore,
the setting of the optimal factor combination in static multi-response problems is directly
determined. Köksoy [
30
] introduces a method for optimizing multi-response problems based
on the mean square error (MSE) criterion and generalized reduced gradient (GRG) algorithm
Inventions 2024,9, 31 3 of 14
for non-linear programming. The methodology is effective when the correlation structure of
the responses does not affect the analysis.
Al-Refaie [
31
] proposed an approach for solving the multi-response problem based on the
quadratic loss function, grey relation analysis, and efficiency technique in data envelopment
analysis (DEA). He treated each experiment in Taguchi’s orthogonal array (OA) as a decision-
making unit (DMU), while the grey relational coefficients were set as the inputs for all DMUs.
Pal and Gauri [
32
] proposed a new multi-response optimization approach using multiple
regression-based weighted Taguchi’s signal-to-noise ratio (MRWSN). The optimal factor-level
combination is obtained considering the weighted signal-to-noise ratio as the overall process
performance index. Al-Refaie et al. [
33
] used fuzzy regression and desirability functions
to solve multi-response problems in the Taguchi method based on the S/N ratio approach.
The optimization model is constructed using the desirability function and the process’s
performance. Canessa et al. [
34
] developed a static robust design for multi-objective problems
based on Taguchi’s parameter design approach and a Pareto genetic algorithm (PGA). This
method is used for designs that are highly fractioned, such as Taguchi’s orthogonal arrays.
The robust design for the multi-objective problems is found in the Pareto frontier of solutions.
Many authors [
35
–
38
] have integrated only the S/N ratio approach and grey relation analysis
to optimize multi-objective quality characteristics. Parinam et al. [
39
] presented a design
parameter optimization procedure, which combines the Taguchi technique with a genetic
algorithm to obtain the optimal values of the design parameters. Parnianifard et al. [
40
]
introduced a novel multi-objective robust optimization method, which investigates the best
levels of the design variables such that a trade-off between robustness, production cost, and
process performance is obtained. The approach is based on response surface methodology,
quality loss function, and process capability ratio.
Viswanathan et al. [
41
] combined the S/N ratio approach and the grey relational
analysis with a principal component analysis based on the response surface methodology to
optimize the optimal parameter-setting process. Sreedharan et al. [
42
] combined a weighted
grey relational analysis with a principal component analysis as well as the desirability
analysis in response surface methodology to obtain the best combination set that optimizes
the process with a multi-objective response. Kumar and Mondal [
43
] applied the technique
for order of preference by similarity to ideal solution (TOPSIS) and grey relational analysis
to investigate the capability of optimizing the output performance characteristics of a
process. Li and Zhu [
44
] and Huang et al. [
45
] applied, in their first work, an intelligent
modeling method by combining three approaches, the S/N ratio approach, the grey relation
analysis, and the fuzzy logic technique, to find the multiple quality characteristic-optimized
process parameters. Lin et al. [
46
] determined the optimal processing conditions for
simultaneously optimizing two responses with conflicting goals by applying a hybrid
approach based on the Taguchi robust design methodology and the grey relation analysis
theory. Jiang and Zou [
47
] proposed a hybrid model to transform multiple Taguchi S/N
ratios into a composite response variable in a way similar to, but slightly different from,
data envelopment analysis. To enable companies to focus on continuous improvement,
Tanash et al. [
48
] implemented a Deming cycle (Plan–Do–Check–Act: PDCA) as a formal
procedure of the improvement approach in a multi-criteria decision-making problem to
ensure the consistency and sustainability of the enhancement methods. They used the
S/N ratio approach and a fuzzy model to produce a single comprehensive output measure
to be optimized. Motivated by the unused potential for a robustness evaluation with the
embodiment function relation and tolerance (EFRT-) model, Li et al. [
49
] proposed an
approach that allows information exchange between the contact and channel approach
(C&C²-A) and the tolerance graph. They explored the missing link between the applicable
robustness criteria and the extended information from the tolerance graphs and the C&C²-A.
Chen et al. [
50
] used stochastic gradient descent to formulate and solve design problems
with distributionally robust optimization (DRO) approaches. They studied the connections
between a class of DRO and the Taguchi method in the context of robust design optimization.
To optimize the parameters of a process with a multi-objective function, Shrimali et al. [
51
]
Inventions 2024,9, 31 4 of 14
proposed a methodology that was based on the Taguchi S/N ratio approach coupled with
a multi-criteria decision-making method, namely the analytic hierarchy process approach.
Zheng et al. [
52
] proposed a new robust design for multi-objective optimization using
probability theory. They took the arithmetic mean values of performance indicators and
deviations as two independent factors to deal with the problem of the robust optimization
of process parameters. The arithmetic mean value of the performance indicator is assessed
as a representative of the performance indicator according to the function, and the deviation
is the other index of the performance indicator, which generally has the characteristic of
“the smaller the better”. For multi-objective optimization, Zheng and Yu [
53
] developed
a robust design approach with an orthogonal experimental methodology in the case of
targeting the best target based on the probabilistic method. The objectives are the difference
between the target value and the arithmetic mean value of performance indicators for the
alternatives and the square root of the mean squared error of actual performance indicator
values from the target value of the alternatives.
Based on the Taguchi approach, Parnianifard et al. [
54
] classified eighteen hybrid
metamodels for robust design optimization. The common goal of these many methods
is the robust design and accurate optimization of the processes. However, the process
environment, uncertainties, uncontrollable factors, and the number of conflicting responses
may bias the optimal response.
This paper is structured as follows. Section 1.2 reviews the robust design of products
and processes using the stochastic frontier method (RDPP-SF) [
13
,
14
]. Section 2presents the
material and methods for the EPD coating process. Section 3discusses the statistical analysis
and main findings of the research. The conclusion (Section 4) presents the main findings
concerning the EPD process. The limitations of the RDPP-SF method and perspective are
also considered.
1.2. Static RDPP-SF Method
SFA stands for stochastic frontier analysis, and it is a commonly used technique in the
econometric field. It is used to model and estimate the technical, allocative, and economic
inefficiency of the decision-making units (DMUs) in the framework of the production
functions [
55
]. Originally, work on the stochastic frontier production functions is traced
back to Aigner et al. [
56
], Meeusen and Van Den Broeck [
57
], and Schmidt and Knox
Lovell [58]. The math model as proposed by Aigner et al. [56] is given in Equation (1).
yi=f(xi;β)+ (vi−ui)i=1, . . . , N (1)
where the observed logged output values y
i
are bound above by exp(
xiβ+vi
); x
i
is vector
of k logged values, x
i
= [1 ln(x
1i
) ln(x
2i
)
. . .
ln(x
ki
)] used by the producer i; and
β
is a vector
of technology parameters to be estimated, β= [β0β1. . . βk]’.
The common assumption for the v
i
and u
i
distributions are the normal and semi-
normal distributions, respectively [59]:
(a)
vi~N(0, σ2
v).
(b)
ui~N+(0, σ2
u).
(c)
viand uiare independent and uncorrelated with the input variables.
The parametric stochastic frontier (SF) model, which is stated in Equation (1), accom-
modates a compound error structure; a neoclassic symmetrical error (v
i
), which represents
the random disturbance in a process or a production unit (e.g., random chocks, mea-
surement error, uncontrolled explanatory variables, etc.); and another positive one-sided
disturbance term (u
i
), which stands for the technical efficiency (TE) of the process of DMUs.
The production DMUs, which operate under the same technology, are expected to reach
the frontier as the maximum attainable output for a given set of input resources (x
i
). However,
in practice, because of operating and managerial circumstances and exogenous variables that
are beyond the control of the DMU, the inefficiency error (u
i
) occurs, explaining why a DMU
cannot achieve the maximum feasible output performance beyond noise variation.
Inventions 2024,9, 31 5 of 14
The static RDPP-SF method, which was originally devised by Trabelsi and Rezgui [
13
]
and amended by Rezgui and Trabelsi [
14
] to consider multi-objective processes, borrowed
from the econometric model to isolate natural (i.e., experimental unit-to-unit) and non-
natural (i.e., environment, use, and deterioration) sources of variation. As for the production
DMUs, in a manufacturing process, the inputs are the operating parameters, and the outputs
are the process responses. In engineering applications, a planned experiment is typically
carried out to determine the optimal combination of the input factor levels, which yields
the least amount of variation in the process response(s) when environmental noises are in
action. Comparable with the econometric field, the RDPP-SF method [
13
,
14
] states that each
combination level of the operating parameters (i.e., each run of the designed experiment)
would yield variation in the process response(s). The variation is composed of a neoclassic
pure random term arising from the sampling strategy and unit-to-unit variation and a
non-random component, which may originate from dynamic functional degradation, tool
and batch error, material contamination, measurement calibration, etc. Robust systems and
processes are sustainable in time, meaning that they are a priori under statistical control.
Therefore, they are only laced with pure random variation. When compared with the
econometric field, the RDPP-SF method matches the robustness to the inefficiency of the
DMUs, which is now estimated for each run of the planned experiment. The lower the u
i
term, the higher the robustness. Table 1shows the mapping scheme between the stochastic
econometric and the RDPP-SF models [13].
Table 1. Mapping table econometric—RDPP-SF.
Econometric Model RDPP-SF Method Performance Metric
DMUs Design plan (DoE)
γ=σ2
u
σ2
v+σ2
u
DMUi Execution i in the DoE
Cross/panel data Nonreplicated/replicated
vi: Random variation Natural noise (experimental and unit-to-unit variation)
ui: TE Non-natural noise (environment and degradation)
Designed experiments using the response surface method (RSM) are usually employed
in optimization problems, especially when curvature is suspected [
60
]. In RSM, the sys-
tem’s probabilistic response is represented by linear or quadratic models. This is sufficient
for most engineering problems, which are investigated using at least an IV resolution
design. The following steps are involved in the RSM procedure: first, a number of the most
influential random variables are chosen; second, the system response is assessed using a
deterministic analysis for each set of values of the chosen random variables; third, using
the data gathered from the deterministic analyses, a linear or quadratic approximation is
constructed to represent the system response by regression analysis; and lastly, the approxi-
mate closed-form representation is obtained, and the system’s probabilistic characteristics
are assessed using contour and surface plots and tools like (MCS). The procedural scheme
of the RDPP-SF method for static multi-objective problems follows the steps shown in
Figure 1[13,14].
Step 1 defines the DoE strategy and assigns factor levels (
±
3
σ
coding is recommended,
while any other coding is also relevant). Each combination of the factors set at each trial
(run of the DoE) is viewed as a decision-making unit (DMU), which uses the sources x
i
[
61
].
Each process response (Y
i
) is transformed and scaled upon the optimization objective,
i.e., nominalization, maximization, or minimization. Because the stochastic production
function initially considers the maximum output that is attainable for a combination of
inputs (x
i
), the outputs of the types smaller-the-best (STB) and nominal-the-best (NTB)
should be transformed to meet maximization and scaled afterwards. This is performed
using the transformation function as in Equations (2) and (3).
Nominal-The-Best (NTB): Yi=exp[−abs(yi−yT)] (2)
Inventions 2024,9, 31 6 of 14
Smaller-The-Best (STB)Yi=1
yi
(3)
Yscaled =(Ub−Lb)Yi−Ymin
YMax −Ymin
+Lb(4)
where y
i
, y
T
, Y
i
, Y
scaled
, U
b
, and L
b
are the original output (not transformed), target value,
transformed output (not scaled), and transformed and scaled output, which are used in the
RDPP-SF method, upper, and lower bound of the original data interval, respectively.
1
Figure 1. Functional scheme for the static multi-objective RDPP-SF method.
To return to the original interval of the raw data for the STB and the NTB performance
characteristics, a scaling procedure is carried out as in Equation (4). Also, it is advised to
analyze the stochastic frontier model using the graded levels of the inputs (x
i
) as opposed
to the engineering units. This is because coded inputs are effective for determining the
relative size of the individual effects of the input parameters; moreover, they allow for
homogenous estimates of the regression coefficients of the frontier model.
Step 2 chooses the transfer function for the stochastic frontier model. A quadratic
translog (TL) model is recommended to account for factor interactions. We execute the
FRONTIER4.1
®
program for each output (Y
i
) and test the following hypotheses at a 95%
confidence level.
a.
H
0
:
βk
= 0 vs. H
1
:
βk=
0. This is to check the statistical significance of the coefficients
of the frontier model.
b.
H
0
:
γ
= 0 vs. H
1
:
γ
> 0. This is to assess the fitness of the average line model (RSM)
and investigate the statistical significance of the special cause error for the (Yi).
c. H0: Ui~Half-normal vs. H1: Ui~truncated normal distribution.
Inventions 2024,9, 31 7 of 14
Only hypotheses (a) and (b) are checked for the RDPP-SF method [
13
,
14
]. The distri-
butional form of the Ui is assumed to be half-normal.
Step 3, the printout, which is generated by the FRONTIER4.1
®
program, provides the
regression model of the transfer function, the
γ
-value for each system response (Y
i
), and
the inefficiency score of each run. The error array for Y
i
is based on the
γ
-value and the
inefficiency score u
i
for each run (i). The u
i
(s) are obtained from the FRONTIER4.1
®
print
output as the individual inefficiency (exp(-ui)).
a.
If
γ≥
95%, then v
i≈
0 and special cause variation prevails. The error array (e
i
) is
then composed of the ui(s) values of the output (Yi) for each run.
b.
If
γ≤
5%, then u
i≈
0 and the bulk of variation is due to only random unit-to-unit
variation. The error array (e
i
) is composed of the vi(s) values of the output (Y
i
) for each
run. In this situation, the average line model (RMS) is confounding with the SF model.
c.
If 5%
≤γ≤
95%, both random and special cause variation sources are accountable
for the result. The error array (e
i
) is then composed of the Abs(v
i
–u
i
) values of the
output (Yi) for each run. The (vi–ui) values represent the observable variation in Yi.
Step 4 is concerned with lessening variation in each Y
i
for the multiple quality char-
acteristics process. The least sensitive solution would correspond to the run (i*) in the
designed experiment, which adds up to a minimum of
(∑abs(ei))
accounting for all Y
i
(s).
2. Materials and Methods
Barbana et al. [
62
] have conducted a central composite design (CCD) as an exper-
imental strategy [
63
] to investigate the effects of four operating factors, i.e., the initial
concentration (x
1
), the deposition time (x
2
), the duty cycle (x
3
), and the voltage (x
3
) on the
properties of the TiO
2
film. The levels of the operating factors (x
i
) are listed in Table 2. The
SS-316L substrate preparation and pulsed electrophoretic deposition process are described
in Barbana et al. [4]. Figure 2shows the experimental coating process.
Inventions 2024, 9, x FOR PEER REVIEW 8 of 15
Figure 2. Schematic illustration of a pulse circuit generator and working electrophoresis cell [4].
The process performance characteristics, i.e., De (photocatalytic efficiency of the thin
film) and three critical charges, LC1, LC2, and LC3, are used to characterize the properties of
the TiO2 films. LC1 is defined as the load at which the first cracks occurred (cohesive fail-
ure); LC2 is the load at which the film starts to delaminate at the edge level of the scratch
track (adhesion failure); and LC3 is the load when the damage of the film exceeds 50%. The
degradation experiments allow us to calculate the decolorization efficiency (De). All re-
sponses should be maximized. Table 3 shows the CCD layout as well as the process re-
sponses (De, LC1, LC2, and LC3). The 30 manipulations are prepared in random order for
homogeneity.
Table 3. CCD layout for the experimental study.
Run Operating Parameters (Coded Values) Performance Characteristics (PCHs)
x1 (g/L) x2 (s) x3 (%) x4 (V) De (%) LC1 (N) LC2 (N) LC3 (N)
1 −1 −1 −1 −1 76.39 4.69 8.88 9.23
2 1 −1 −1 −1 83.57 3.89 7.56 9.56
3 −1 1 −1 −1 68.77 4.67 7.32 10.74
4 1 1 −1 −1 58.87 2.32 6.89 13.84
5 −1 −1 1 −1 57.58 4.78 9.29 11.21
6 1 −1 1 −1 61.96 3.24 7.52 11.47
7 −1 1 1 −1 31.82 5.19 10.05 10.78
8 1 1 1 −1 29.23 2.92 6.59 12.85
9 −1 −1 −1 1 30.96 3.61 6.53 9.64
10 1 −1 −1 1 38.36 4.39 7.94 8.96
11 −1 1 −1 1 43.90 2.56 5.32 10.92
12 1 1 −1 1 41.68 3.25 6.36 10.45
13 −1 −1 1 1 42.26 2.97 5.51 11.85
14 1 −1 1 1 50.09 3.80 6.25 9.35
15 −1 1 1 1 34.42 2.94 4.48 10.30
16 1 1 1 1 34.62 2.75 4.35 10.26
17 0 0 0 0 52.09 3.46 6.35 9.56
18 0 0 0 0 51.89 3.56 7.34 9.17
Figure 2. Schematic illustration of a pulse circuit generator and working electrophoresis cell [4].
The process performance characteristics, i.e., De (photocatalytic efficiency of the thin film)
and three critical charges, L
C1
, L
C2
, and L
C3
, are used to characterize the properties of the TiO
2
films. L
C1
is defined as the load at which the first cracks occurred (cohesive failure); L
C2
is
the load at which the film starts to delaminate at the edge level of the scratch track (adhesion
failure); and L
C3
is the load when the damage of the film exceeds 50%. The degradation
experiments allow us to calculate the decolorization efficiency (De). All responses should be
Inventions 2024,9, 31 8 of 14
maximized. Table 3shows the CCD layout as well as the process responses (De, L
C1
, L
C2
, and
LC3). The 30 manipulations are prepared in random order for homogeneity.
Table 2. The setting of the engineering factors’ levels used in the CCD plan.
Operating Factors Units
Levels
−α−1 0 +1 +α
x1: Initial concentration g/L 2 8 14 20 26
x2: Deposition time s 150 300 450 600 750
x3: Duty cycle (DC) % 10 30 50 70 90
x4: Voltage V 4 22 40 58 76
Table 3. CCD layout for the experimental study.
Run
Operating Parameters (Coded Values) Performance Characteristics (PCHs)
x1(g/L) x2(s) x3(%) x4(V) De (%) LC1 (N) LC2 (N) LC3 (N)
1−1−1−1−1 76.39 4.69 8.88 9.23
2 1 −1−1−1 83.57 3.89 7.56 9.56
3−1 1 −1−1 68.77 4.67 7.32 10.74
4 1 1 −1−1 58.87 2.32 6.89 13.84
5−1−1 1 −1 57.58 4.78 9.29 11.21
6 1 −1 1 −1 61.96 3.24 7.52 11.47
7−1 1 1 −1 31.82 5.19 10.05 10.78
8 1 1 1 −1 29.23 2.92 6.59 12.85
9−1−1−1 1 30.96 3.61 6.53 9.64
10 1 −1−1 1 38.36 4.39 7.94 8.96
11 −1 1 −1 1 43.90 2.56 5.32 10.92
12 1 1 −1 1 41.68 3.25 6.36 10.45
13 −1−1 1 1 42.26 2.97 5.51 11.85
14 1 −1 1 1 50.09 3.80 6.25 9.35
15 −1 1 1 1 34.42 2.94 4.48 10.30
16 1 1 1 1 34.62 2.75 4.35 10.26
17 0 0 0 0 52.09 3.46 6.35 9.56
18 0 0 0 0 51.89 3.56 7.34 9.17
19 0 0 0 0 50.57 3.55 6.14 9.75
20 0 0 0 0 53.24 3.87 6.71 9.64
21 0 0 0 0 55.09 3.52 5.98 9.43
22 0 0 0 0 51.84 3.77 6.85 9.58
23 −α0 0 0 36.13 3.50 5.26 8.41
24 +α0 0 0 35.52 2.60 4.97 8.53
25 0 −α0 0 60.96 5.89 8.92 9.30
26 0 +α0 0 41.31 3.96 6.44 10.33
27 0 0 −α0 70.95 3.24 6.92 10.45
28 0 0 +α0 45.37 3.06 6.911 11.94
29 0 0 0 −α57.62 4.86 9.96 14.46
30 0 0 0 +α24.56 2.92 5.98 12.05
3. Results and Discussion
The four steps listed above in Section 4are followed when implementing the RDPP-SF
method for static systems.
Step 1_ Data preparation: The data are arranged as required by the FRONTIER 4.1
®
program. Because the process outputs De, L
C1
, L
C2
, and L
C3
are all larger-the-best types,
no transformation is needed to satisfy the requirements of the stochastic frontier model.
Step 2_ Execution of FRONTIER 4.1
®
program and hypotheses testing: We choose a
5% type I error and a translog as a transfer function for the process responses. This choice
is supported by the log-ratio values shown in Table 4(tests 1); i.e., LR-stat = 37.98 for De,
Inventions 2024,9, 31 9 of 14
50.94 for L
C1
, 48.92 for L
C2
, and 27.36 for L
C3
, respectively. The coefficients of the regression
models for De, LC1, LC2, and LC3 are shown in Table 5.
Table 4. Hypotheses tests on the SF models for the De, LC1, LC2, and LC3 PCHs outputs.
Hypotheses (α= 5%) LR-stat. χ20.95-Value Decision
De Test 1 Linear vs. quadratic 37.98 18.31 Reject
Test 2 γ= 0 13.67 2.71 Reject
LC1 Test 1 Linear vs. quadratic 50.94 18.31 Reject
Test 2 γ= 0 0.00 2.71 Fail to Reject
LC2 Test 1 Linear vs. quadratic. 48.92 18.31 Reject
Test 2 γ= 0 42.40 2.71 Reject
LC3 Test 1 Linear vs. quadratic. 27.36 18.31 Reject
Test 2 γ= 0 0.00 2.71 Fail to reject
Table 5. Estimates of the SF models for De, LC1, LC2, and LC3 outputs.
Variables Param.
De(%)
(γ=0; µ=η= 0)
LC1(N)
(γ=0; µ=η= 0)
LC2(N)
(γ= 0, OLS)
LC3(N)
(γ= 0, OLS)
Est. t-test. Est. t-test. Est. t-test. Est. t-test.
Cte. β0−1.244 −1.269 14.016 14.016 −4.185 4.280 −3.777 1.579
ln(x1)β12.409 2.578 * 0.703 0.703 0.685 0.730 −0.019 −0.026
ln(x2)β24.070 6.575 * −3.508 −3.508 * 0.578 1.082 0.385 0.586
ln(x3)β31.679 1.981 −0.613 −0.613 1.419 1.645 1.512 1.719
ln(x4)β4−6.007 −6.858 * −0.284 −0.284 1.139 1.378 0.875 1.321
ln(x1)ˆ2 β5−0.153 −3.602 * −0.098 −0.098 −0.141 −4.960 * −0.072 −2.536
ln(x
1
)*ln(x
2
)
β6−0.294 −1.804 −0.373 −0.373 −0.092 −0.758 0.234 2.267
ln(x
1
)*ln(x
3
)
β70.010 0.061 −0.138 −0.138 −0.230 −2.259 * −0.063 −0.763
ln(x
1
)*ln(x
4
)
β80.020 0.138 0.664 0.664 0.383 4.390* −0.224 −3.027
ln(x2)ˆ2 β9−0.290 −3.644 * 0.279 0.279 −0.011 −0.196 0.063 1.014
ln(x
2
)*Ln(x
3
)
β10 −0.670 −5.869 * 0.362 0.362 0.055 0.430 −0.315 −2.706
ln(x
2
)*Ln(x
4
)
β11 0.666 4.660 * −0.144 −0.144 −0.166 −1.549 −0.114 −1.215
ln(x3)ˆ2 β12 −0.047 −1.110 −0.080 −0.080 −0.031 −0.724 0.084 2.152
ln(x
3
)*ln(x
4
)
β13 0.682 4.155 * −0.185 −0.185 −0.258 −2.916 * 0.002 0.027
ln(x4)ˆ2 β14 −0.159 −4.281 * −0.002 −0.002 −0.055 −2.265 * 0.046 2.161
σ2=σ2
v+σ2
u0.03 0.005 0.012 0.005
γ-value 1.000 0.05 1.000 0.000
µ,η----
Log (likelihood) 31.79 36.48 42.40 38.44
Critical t-value (5%) = 1.753
* Starred coefficients are significant parameters at a 5% level.
Table 5indicates that assuming a 5% error level, the
γ
-values for the L
C1
and L
C3
pro-
cess responses are inferior to 5%, meaning that most of the residual variation is attributable
to pure random sources of variation (sampling and part-to-part process variation). There-
fore, the SF and neoclassic response surface model (RSM), are confounding. However, the
high
γ
-values of 1 (at 3 dp) for the process outputs De and L
C2
indicate that the bulk of
variation is due to non-natural sources and that the SF and RSM are not confounding.
Step 3_ Constitute the error array for each process response (Y
i
): As for the
γ
-values,
the abs(v
i
) terms at each run of the CCD layout make up the error arrays for the L
C1
and
L
C3
outputs, while the u
i
values make up the error arrays for the De and L
C2
outputs.
Table 6shows the error matrix for the De, LC1, LC2, and LC3 process responses.
Step 4_ Determination of the robust design solution: The error terms are added over
the process responses for every run (i) in the CCD layout. The robust design solution
corresponds to the one having a minimum value. Allowing for a 95% confidence interval,
Table 6and Figure 3suggest that run 25 (initial concentration of 14 g/L, deposition time
of 150 s, duty cycle of 50%, and voltage of 40 V) is the robust optimization setting for
the pulsed electrophoretic deposition parameters on TiO
2
film properties. Other potential
design solutions, such as runs 23 and 29, should also be investigated based on their
functionality, quality, and cost. Irrespectively, at a duty cycle of 50%, the voltage is the
Inventions 2024,9, 31 10 of 14
best sitting, which does not affect the sensitivity of the four outputs, De, L
C1
, L
C2
, and L
C3
.
The level combinations of runs 24 and 30 yield the most sensitive design solutions, so they
should be avoided at any cost.
Table 6. Global error table for the De, LC1, LC2, and LC3 outputs.
Run
Operating Factors PCHs
Global Error Ranking
x
1
(g/L)
x2(s) x3(%) x4(V) De(%) LC1(N) LC2(N) LC3(N)
uiAbs(vi) uiAbs(vi)
1 8 300 30 22 0.253 0.066 0.004 0.038 0.360 21
2 20 300 30 22 0.209 0.055 0.021 0.036 0.321 17
3 8 600 30 22 0.171 0.014 0.144 0.040 0.368 23
4 20 600 30 22 0.185 0.136 0.002 0.108 0.431 27
5 8 300 70 22 0.220 0.032 0.143 0.032 0.426 26
6 20 300 70 22 0.199 0.004 0.032 0.067 0.302 14
7 8 600 70 22 0.232 0.076 0.043 0.028 0.379 24
8 20 600 70 22 0.183 0.001 0.084 0.067 0.335 19
9 8 300 30 58 0.202 0.025 0.040 0.011 0.278 11
10 20 300 30 58 0.050 0.059 0.041 0.078 0.228 9
11 8 600 30 58 0.113 0.134 0.081 0.040 0.368 22
12 20 600 30 58 0.041 0.057 0.040 0.010 0.148 4
13 8 300 70 58 0.134 0.002 0.182 0.007 0.326 18
14 20 300 70 58 0.035 0.070 0.074 0.033 0.211 8
15 8 600 70 58 0.206 0.043 0.257 0.035 0.541 28
16 20 600 70 58 0.084 0.046 0.245 0.023 0.399 25
17 14 450 50 40 0.056 0.012 0.145 0.065 0.278 12
18 14 450 50 40 0.060 0.040 0.000 0.107 0.207 7
19 14 450 50 40 0.086 0.037 0.179 0.046 0.347 20
20 14 450 50 40 0.034 0.122 0.090 0.057 0.303 15
21 14 450 50 40 0.000 0.029 0.205 0.079 0.313 16
22 14 450 50 40 0.061 0.096 0.069 0.063 0.289 13
23 2 450 50 40 0.005 0.040 0.016 0.010 0.070 2
24 26 450 50 40 0.328 0.121 0.267 0.123 0.839 30
25 14 150 50 40 0.003 0.011 0.012 0.032 0.057 1
26 14 750 50 40 0.001 0.173 0.025 0.068 0.268 10
27 14 450 10 40 0.013 0.036 0.057 0.050 0.155 5
28 14 450 90 40 0.037 0.039 0.021 0.075 0.172 6
29 14 450 50 4 0.014 0.025 0.042 0.044 0.125 3
30 14 450 50 76 0.491 0.062 0.005 0.189 0.747 29
Inventions 2024, 9, x FOR PEER REVIEW 11 of 15
25 14 150 50 40 0.003 0.011 0.012 0.032 0.057 1
26 14 750 50 40 0.001 0.173 0.025 0.068 0.268 10
27 14 450 10 40 0.013 0.036 0.057 0.050 0.155 5
28 14 450 90 40 0.037 0.039 0.021 0.075 0.172 6
29 14 450 50 4 0.014 0.025 0.042 0.044 0.125 3
30 14 450 50 76 0.491 0.062 0.005 0.189 0.747 29
Step 4_ Determination of the robust design solution: The error terms are added over
the process responses for every run (i) in the CCD layout. The robust design solution cor-
responds to the one having a minimum value. Allowing for a 95% confidence interval,
Table 6 and Figure 3 suggest that run 25 (initial concentration of 14 g/L, deposition time
of 150 s, duty cycle of 50%, and voltage of 40 V) is the robust optimization setting for the
pulsed electrophoretic deposition parameters on TiO2 film properties. Other potential de-
sign solutions, such as runs 23 and 29, should also be investigated based on their func-
tionality, quality, and cost. Irrespectively, at a duty cycle of 50%, the voltage is the best
sitting, which does not affect the sensitivity of the four outputs, De, LC1, LC2, and LC3. The
level combinations of runs 24 and 30 yield the most sensitive design solutions, so they
should be avoided at any cost.
Figure 3. Global error values at each run of the CCD plan.
4. Conclusions
The static RDPP-SF method was used for the robust design optimization of the elec-
trophoretic deposition process on TiO2 films. The results of the hypothesis test regarding
the residual variation in the De and LC2 responses show significance at the 5% level (both
responses have a γ-value of one). Consequently, as the overall variation in the De and LC2
is not random, the EDP is not under statistical control concerning these outputs. Further-
more, the stochastic frontier and RSM are not confounding. From an output point of view
of the stochastic production function, higher levels of De and LC2 outputs could be ob-
tained while using the same input resources (xi). For the LC1 and LC3 outputs, the signifi-
cance tests on non-random variation are insignificant at a 5% level (γ-values equals 0.05
and 0.00, respectively) meaning that the variation in the LC1 and LC3 process responses is
only due to pure random sources (part-to-part variation). As a result, the least square re-
sponse model (RSM) coincides with the stochastic frontier.
Table 6 shows the added values for the error arrays regarding the outputs De and the
three critical charges LC1, LC2, and LC3. Accordingly, the robust optimization design for the
Figure 3. Global error values at each run of the CCD plan.
Inventions 2024,9, 31 11 of 14
4. Conclusions
The static RDPP-SF method was used for the robust design optimization of the elec-
trophoretic deposition process on TiO
2
films. The results of the hypothesis test regarding
the residual variation in the De and L
C2
responses show significance at the 5% level (both
responses have a
γ
-value of one). Consequently, as the overall variation in the De and L
C2
is
not random, the EDP is not under statistical control concerning these outputs. Furthermore,
the stochastic frontier and RSM are not confounding. From an output point of view of
the stochastic production function, higher levels of De and L
C2
outputs could be obtained
while using the same input resources (x
i
). For the L
C1
and L
C3
outputs, the significance
tests on non-random variation are insignificant at a 5% level (
γ
-values equals 0.05 and 0.00,
respectively) meaning that the variation in the L
C1
and L
C3
process responses is only due to
pure random sources (part-to-part variation). As a result, the least square response model
(RSM) coincides with the stochastic frontier.
Table 6shows the added values for the error arrays regarding the outputs De and the
three critical charges L
C1
, L
C2
, and L
C3
. Accordingly, the robust optimization design for the
pulsed electrophoretic deposition parameters corresponds to run 25 (minimum total). The
robust optimization solution corresponds to the following setting: x
1
: initial concentration
at 14 g/L; x
2
: deposition time at 150 s; x
3
: duty cycle at 50%; and x
4
: applied voltage at
40. Runs 23 and 29 are additional potential runs that should be accounted for from an
economic standpoint. This study also showed that the duty cycle of 50% voltage is the best
setting, which does not affect the sensitivity of the four outputs, De, L
C1
, L
C2
, and L
C3
(see
Tables 6and 7).
Table 7. Optimization results of RDPP-SF [13,14] vs. RS-DF [62] methods.
Operating Factors Process Responses
x1(g/L) x2(s) x3(%) x4(V) De (%) LC1 (N) LC2 (N) LC3 (N)
γ-Value - - - - 1.000 0.050 1.000 0.000
RDPP-SF
run 25 14 150 50 40 60.960 5.890 8.920 9.300
run 23 2 450 50 40 36.130 3.500 5.260 8.410
run 29 14 450 50 4 57.620 4.860 9.960 14.460
RSM-Desirability [41] method 16.34 150 90 4 82.757 5.895 12.584 16.773
Using the response surface model (RSM) and desirability function method (RS-DF) as
suggested in Barbana et al. [62], the optimum in continue space is obtained when setting the
concentration of TiO
2
at 16.34 g/L, the deposition time at 150 s, the duty cycle at 90%, and the
applied voltage at 4 V. The discrepancy with the RDPP-SF method is partly due to the fact the
RS-DF method [
62
] does not account for the non-random noise sources in the De% and L
C2
responses. According to Table 7, pure non-natural variation (
γ
-value of nearly one) is present
in both process responses. This suggests that contrary to Barbana et al. [
62
], the RSM should
not be used to estimate the optimal setting for the De% and LC2 responses.
For the robust optimization of the EPD process, the RDPP-SF method can be viewed
as a long-term reliability approach where part-to-part variation is compounded with non-
natural sources of variation (noise variables). The objective is to engineer the process mean
square deviation (MSD), which is associated with long-term process part-to-part, use, and
deteriorative sources of variation.
The signal-to-noise metric, which is employed in many robust design techniques, can
be linked with the RDPP-SF approach using the estimation of the
γ
-value metric. Thus, in
engineering domains like reliability, resilience, adaptability, and versatility, the RDPP-SF
approach has the greatest potential. However, the RDPP-SF method suffers from two major
limitations: (i) A translog transfer function is used in the RDPP-SF method to consider
the interactions between the control factors. Further investigation is required to identify
alternative functional forms for engineering processes, and (ii) The RDPP-SF method uses
graded rather than continued scales to code process parameters. An investigation utilizing
Inventions 2024,9, 31 12 of 14
a hybrid ANN-GA framework is being conducted to address the second limitation of the
RDPP-SF approach.
Adopting optimized electrophoretic deposition (EPD) parameters in industrial set-
tings offers substantial economic advantages. By refining these parameters, companies
can achieve cost savings through enhanced process efficiency and minimized material
wastage. Improved product quality resulting from optimized EPD parameters leads to
heightened customer satisfaction and loyalty. Furthermore, the scalability of the process
enables companies to handle larger production volumes without proportional increases in
costs. Overall, these optimizations bolster industry competitiveness and profitability by
streamlining operations and maximizing resource utilization.
Author Contributions: Conceptualization, A.T., M.A.R. and N.B.; methodology, A.T., M.A.R., N.B.
and A.B.Y.; validation, N.B. and A.B.Y.; investigation, M.A.R. and N.B.; data curation, A.T. and N.B.;
writing—original draft preparation, A.T., M.A.R. and N.B.; writing—review and editing, M.A.R., N.B.
and M.A.-A. supervision, M.A.-A. All authors have read and agreed to the published version of the
manuscript.
Funding: This research received no external funding.
Data Availability Statement: The data that support the findings of this study are available from the
corresponding author (Nesrine Barbana) upon reasonable request.
Acknowledgments: We acknowledge support from the German Research Foundation and the Open
Access Publication Fund of the Technical University of Berlin.
Conflicts of Interest: The authors declare no potential conflicts of interest concerning the research,
authorship, and/or publication of this article.
References
1.
Zeribi, F.; Attaf, A.; Derbali, A.; Saidi, H.; Benmebrouk, L.; Aida, M.S.; Dahnoun, M.; Nouadji, R.; Ezzaouia, H. Dependence of the
physical properties of titanium dioxide (TiO
2
) thin films grown by sol-gel (spin-coating) process on thickness. ECS J. Solid State
Sci. Technol. 2022,11, 023003. [CrossRef]
2.
Xu, J.; Nagasawa, H.; Kanezashi, M.; Tsuru, T. TiO
2
coatings via atmospheric-pressure plasma-enhanced chemical vapor
deposition for enhancing the UV-resistant properties of transparent plastics. ACS Omega 2021,6, 1370–1377. [CrossRef]
3.
Puga, M.L.; Venturini, J.; Ten Caten, C.S.; Bergmann, C.P. Influencing parameters in the electrochemical anodization of TiO
2
nanotubes: Systematic review and meta-analysis. Ceram. Int. 2022,48, 19513–19526. [CrossRef]
4.
Barbana, N.; Ben Youssef, A.; Dhiflaoui, H.; Bousselmi, L. Preparation and characterization of photocatalytic TiO
2
films on
functionalized stainless steel. J. Mater. Sci. 2018,53, 3341–3364. [CrossRef]
5.
Ben Youssef, A.; Barbana, N.; Al-Addous, M.; Bousselmi, L. Preparation and characterization of photocatalytic TiO
2
/WO
3
films
on functionalized stainless steel. J. Mater. Sci. Mater. Electron. 2018,29, 19909–19922. [CrossRef]
6.
Santillán, M.; Membrives, F.; Quaranta, N.; Boccaccini, A. Characterization of TiO
2
nanoparticle suspensions for electrophoretic
deposition. J. Nanopart. Res. 2008,10, 787–793. [CrossRef]
7.
Hanaor, D.; Michelazzi, M.; Veronesi, P.; Leonelli, C.; Romagnoli, M.; Sorrell, C. Anodic aqueous electrophoretic deposition of
titanium dioxide using carboxylic acids as dispersing agents. J. Eur. Ceram. Soc. 2011,31, 1041–1047. [CrossRef]
8.
de Sousa, R.R.M.; de Araújo, F.O.; da Costa, J.A.P.; Nishimoto, A.; Viana, B.C.; Alves, C., Jr. Deposition of TiO
2
film on duplex
stainless-steel substrate using the cathodic cage plasma technique. Mater. Res. 2016,19, 1207–1212. [CrossRef]
9.
Gao, F.; Sherwood, P. Photoelectron spectroscopic studies of the formation of hydroxyapatite films on 316L. Surf. Interface Anal.
2012,44, 1587–1600. [CrossRef]
10.
Krishna, D.; Chena, Y.; Sun, Z. Magnetron sputtered TiO
2
films on a stainless-steel substrate: Selective rutile phase formation and
its tribological and anti-corrosion performance. Thin Solid Films 2011,519, 4860–4864. [CrossRef]
11.
Tang, F.; Uchikoshi, T.; Ozawa, K.; Sakka, Y. Effect of polyethyleneimine on the dispersion and electrophoretic deposition of
nano-sized titania aqueous suspensions. J. Eur. Ceram. Soc. 2006,26, 1555–1560. [CrossRef]
12.
Nurhayati, E.; Yang, H.; Chen, C.; Liu, C.; Juang, Y.; Huang, C.; Hu, C. Electro-photocatalytic fenton decolorization of orange G
using mesoporous TiO
2
/stainless steel mesh photo-electrode prepared by the sol-gel dip-coating method. Int. J. Electrochem. Sci.
2016,11, 3615–3632. [CrossRef]
13.
Trabelsi, A.; Rezgui, M.-A. Robust design of processes and products using the mathematics of the stochastic frontier. Int. J. Adv.
Manuf. Technol. 2020,106, 2829–2841. [CrossRef]
14.
Rezgui, M.-A.; Trabelsi, A. Use of the stochastic frontier and the grey relational analysis in robust design of multi-objective
problems. Concurr. Eng. Res. Appl. 2020,28, 110–129. [CrossRef]
15. Taguchi, G. System of Experimental Design; UNIPUB/Kraus International Publications: White Plains, NY, USA, 1987.
Inventions 2024,9, 31 13 of 14
16.
Del Castillo, E.; Montgomery, D.; McCarville, D.R. Modified desirability functions for multiple response optimization. J. Qual.
Technol. 1996,28, 337–345. [CrossRef]
17. Chen, L.H. Designing robust products with multiple quality characteristics. Comput. Oper. Res. 1997,24, 937–944. [CrossRef]
18.
Su, C.T.; Tong, L.I. Multi-response robust design by principal component analysis. Total Qual. Manag. 1997,8, 409–416. [CrossRef]
19.
Tong, L.-I.; Su, C.T. Optimizing multi-response problems in the Taguchi method by fuzzy multiple attribute decision making.
Qual. Reliab. Eng. Int. 1997,13, 25–34. [CrossRef]
20.
Antony, J. Simultaneous optimization of multiple quality characteristics in manufacturing processes using Taguchi’s quality loss
function. Int. J. Adv. Manuf. Technol. 2001,17, 134–138. [CrossRef]
21.
Kim, K.J.; Lin, D.K.J. Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability
functions. J. R. Stat. Soc. Ser. C Appl. Stat. 2000,49, 311–325. [CrossRef]
22.
Liao, H.-C.; Chen, Y.K. Optimizing multi-response problem in the Taguchi method by DEA-based ranking method. Int. J. Qual.
Reliab. Manag. 2002,19, 825–837. [CrossRef]
23.
Liao, H.-C. Using PCR-TOPSIS to optimize Taguchi’s multi-response problem. Int. J. Adv. Manuf. Technol. 2003,22, 649–655.
[CrossRef]
24.
Hsu, C.M.; Su, C.T.; Liao, D. Simultaneous of the broadband tap coupler optical performance based on neural networks and
exponential desirability functions. Int. J. Adv. Manuf. Technol. 2004,23, 896–902. [CrossRef]
25.
Ortiz, F.J.R.; Simpson, J.R.; Pignatiello, J.R.; Alejandro, H.L. A genetic algorithm approach to multiple-response optimization. J.
Qual. Technol. 2004,36, 432–449. [CrossRef]
26. Liao, H.-C. Using N-D method to solve multi-response problem in Taguchi. J. Intell. Manuf. 2005,16, 331–347. [CrossRef]
27.
Fung, C.P.; Kang, P.C. Multi-response optimization in friction properties of PBT composites using Taguchi method and principle
component analysis. J. Mater. Process. Technol. 2005,170, 602–610. [CrossRef]
28.
Liao, H.-C. Multi-response optimization using weighted principal component. Int. J. Adv. Manuf. Technol. 2006,27, 720–725.
[CrossRef]
29.
Liao, H.-C.; Chang, H.H.; Hsu, C.M. Using canonical correlation to optimize Taguchi’s multiresponse problem. Concurr. Eng. Res.
Appl. 2006,14, 141–149. [CrossRef]
30.
Köksoy, O.A. Nonlinear programming solution to robust multi-response quality problem. Appl. Math. Comput. 2008,196, 603–612.
[CrossRef]
31.
Al-Refaie, A. Grey-data envelopment analysis approach for solving the multi-response problem in the Taguchi method. Proc.
IMechE Part B J. Eng. Manuf. 2010,224, 147–158. [CrossRef]
32.
Pal, S.; Gauri, S.K. Multi-response optimization using multiple regression-based weighted signal-to-noise ratio (MRWSN). Qual.
Eng. 2010,22, 336–350. [CrossRef]
33.
Al-Refaie, A.; Rawabdeh, I.; Abu-Alhaj, R.; Jalham, I. A fuzzy multiple regressions approach for optimizing multiple responses in
the Taguchi method. Int. J. Fuzzy Syst. Appl. 2012,2, 13–34. [CrossRef]
34.
Canessa, E.; Bielenberg, G.; Allende, H. Robust design in multiobjective systems using Taguchi’s parameter design approach and
Pareto genetic algorithm. Rev. Fac. Ing. Univ. Antioq. 2014,72, 73–85. [CrossRef]
35.
Sylajakumari, P.A.; Ramakrishnasamy, R.; Palaniappan, G. Taguchi grey relational analysis for multi-response optimization of
wear in co-continuous composite. Materials 2018,11, 1743. [CrossRef] [PubMed]
36.
Sutono, S.B. Grey-based Taguchi method to optimize the multi-response design of product form design. J. Optimasi Sist. Ind. 2021,
20, 136–146. [CrossRef]
37.
Huang, W.-T.; Tasi, Z.-Y.; Ho, W.-H.; Chou, J.-H. Integrating Taguchi method and gray relational analysis for auto locks by using
multiobjective design in computer-aided engineering. Polymers 2022,14, 644. [CrossRef]
38. Muthana, S.A.; Ku-Mahamud, K.R. Taguchi-Grey relational analysis method for parameter tuning of multi-objective pareto ant
colony system algorithm. J. Inf. Commun. Technol. 2023,22, 149–181. [CrossRef]
39.
Parinam, S.; Kumar, M.; Kumari, N.; Karar, V.; Sharma, A.L. An improved optical parameter optimisation approach using taguchi
and genetic algorithm for high transmission optical filter design. Optik 2019,182, 382–392. [CrossRef]
40.
Parnianifard, A.; Azfanizam, A.S.; Ariffin, M.K.A.; Ismail, M.I.S. Trade-off in robustness, cost and performance by a multi-objective
robust production optimization method. Int. J. Ind. Eng. Comput. 2019,10, 133–148. [CrossRef]
41.
Viswanathan, R.; Ramesh, S.; Maniraj, S.; Subburam, V. Measurement and Multiresponse Optimization of Turning Parameters for
Magnesium Alloy Using Hybrid Combination of Taguchi GRA- PCA Technique. Measurement 2020,159, 107800. [CrossRef]
42.
Sreedharan, J.; Jeevanantham, A.K.; Rajeshkannan, A. Multi-objective optimization for multi-stage sequential plastic injection
molding with plating process using RSM and PCA-based weighted-GRA. Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 2020,234,
1014–1030. [CrossRef]
43.
Kumar, D.; Mondal, M. Process parameters optimization of AISI M2 steel in EDM using Taguchi based TOPSIS and GRA. Mater.
Today-Proc. 2020,26, 2477–2484. [CrossRef]
44.
Li, Y.; Zhu, L. Multi-objective optimisation of user experience in mobile application design via a grey-fuzzy-based Taguchi
approach. Concurr. Eng.-Res. A 2020,28, 175–188. [CrossRef]
45.
Huang, W.-T.; Tsai, C.-L.; Ho, W.-H.; Chou, J.-H. Application of intelligent modeling method to optimize the multiple quality
characteristics of the injection molding process of automobile lock parts. Polymers 2021,13, 2515. [CrossRef]
Inventions 2024,9, 31 14 of 14
46. Lin, C.-M.; Hung, Y.-T.; Tan, C.-M. Hybrid Taguchi–gray relation analysis method for design of metal powder injection-molded
artificial knee joints with optimal powder concentration and volume shrinkage. Polymers 2021,13, 865. [CrossRef]
47.
Jiang, R.; Zou, B. A DEA-based multi-response fusion model in the context of Taguchi method. In Proceedings of the Fourth
International Conference on Mechanical, Electric and Industrial Engineering, Kunming, China, 22–24 May 2021; Volume 1983,
p. 012108.
48.
Tanash, M.; Al Athamneh, R.; Bani Hani, D.; Rababah, M.; Albataineh, Z. A PDCA framework towards a multi-response
optimization of process parameters based on Taguchi-fuzzy model. Processes 2022,10, 1894. [CrossRef]
49.
Li, J.; Horber, D.; Keller, C.; Grauberger, P.; Goetz, S.; Wartzack, S.; Matthiesen, S. Utilizing the embodiment function relation
and tolerance model for robust concept design. In Proceedings of the International Conference On Engineering Design, ICED23,
Bordeaux, France, 24–28 July 2023.
50.
Chen, L.; Rottmayer, J.; Kusch, L.; Gauger, N.R.; Ye, Y. Data-driven aerodynamic shape design with distributionally robust
optimization approaches. arXiv 2023, arXiv:2310.08931v1.
51.
Shrimali, R.; Kumar, M.; Pandey, S.; Sharma, V.; Kaushik, L.; Singh, K. A robust Taguchi combined AHP approach for optimizing
AISI1023 low carbon steel weldments in the SAW process. Int. J. Interact. Des. Manuf. 2023,17, 1959–1977. [CrossRef]
52.
Zheng, M.; Teng, H.; Wang, Y. Application of new robust design by means of probability-based multi-objective optimization to
machining process parameters. Vojnoteh. Glas. Mil. Tech. Cour. 2023,71, 84–99. [CrossRef]
53.
Zheng, M.; Yu, J. Probabilistic approach for robust design with orthogonal experimental methodology in case of target the best. J.
Umm Al-Qura Univ. Eng. Archit. 2024,15, 55–59. [CrossRef]
54.
Parnianifard, A.; Azfanizam, A.S.; Ariffin, M.K.A.; Ismail, M.I.S. An overview on robust design hybrid metamodeling: Advanced
methodology in process optimization under uncertainty. Int. J. Ind. Eng. Comput. 2018,9, 1–32. [CrossRef]
55.
Coelli, J.T.; Rao, P.; Christopher, O.J.; Battese, E.G. An Introduction to Efficiency and Productivity Analysis, 2nd ed.; Springer: New
York, NY, USA, 2005.
56.
Aigner, D.; Knox Lovell, C.A.; Peter, S. Formulation and estimation of stochastic frontier production function models. J. Econ.
1977,6, 21–37. [CrossRef]
57.
Meeusen, W.; Van Den Broeck, J. Efficiency estimation from Cobb–Douglas production functions with composed error. Int. Econ.
Rev. 1977,18, 435–444. [CrossRef]
58.
Schmidt, P.; Knox Lovell, C.A. Estimating stochastic production and cost frontiers when technical and allocative inefficiency are
correlated. J. Econ. 1980,13, 83–100. [CrossRef]
59.
Von Hirschhausen, C.R.; Cullmann, A.; Kappeler, A. Efficiency analysis of German electricity distribution utilities–non–parametric
and parametric tests. Appl. Econ. 2006,38, 2553–2566. [CrossRef]
60.
Montgomery, D.C. Design and Analysis of Experiments, 9th ed.; John Wiley & Sons, Inc.: Hoboken, NJ, USA; Arizona State
University: Tempe, AZ, USA, 2017; p. 749.
61.
Coelli, J.T. A Guide to FRONTIER Version 4.1: A Computer Program for Stochastic Frontier Production and Cost Function Estimation;
Centre for Efficiency and Productivity Analysis, University of New England: Armidale, Australia, 1996.
62.
Barbana, N.; Ben Youssef, A.; Rezgui, M.-A.; Bousselmi, L.; Al-Addous, M. Modelling, analysis, and optimization of the effects of
pulsed electrophoretic deposition parameters on TiO
2
films properties using desirability optimization methodology. Materials
2020,13, 5160. [CrossRef]
63.
Box, G.E.P.; Wilson, K.B. On the experimental attainment of optimum conditions. J. R. Stat. Soc. Ser. B 1951,13, 1–45. [CrossRef]
Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual
author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to
people or property resulting from any ideas, methods, instructions or products referred to in the content.
