processes

Article
Shewhart Control Charts Implementation for Quality and
Production Management
Marcela Malindzakova 1 , Katarína Čulková 2, * and Jarmila Trpčevská 2

1 Faculty BERG, Technical University of Košice, 042 00 Košice, Slovakia

2 Faculty MMR, Technical University of Košice, 042 00 Košice, Slovakia
* Correspondence: katarina.culkova@tuke.sk; Tel.: +421-55-602-3116

Abstract: Shewhart control charts are suitable for stable but repetitive production processes used
for the subsequent identification of random deviations while indicating breached quality limits.
They provide information on process variability and, at the same time, make it possible to obtain
information on the reliability of monitored processes. The objective of this paper is to assess the quality
characteristics of plastic mouldings for the needs of the automotive industry with the application of
the control charts method, specifically Shewhart control charts. The Shewhart control charts were
applied to evaluate the quality characteristics, or, more specifically, to evaluate the measured width
and length of the produced plastic mouldings by statistical analysis. Statistical analyses show that
the set parameters are not met in the first two days of the test series. An improvement in the process
is observed on the last day of the test series. The process is well set, as confirmed by our verification
of the stability of the process. An important condition for setting the control charts is to observe the
correct chronological arrangement and regular acquisition of measured values. Solving tasks in the
future must be oriented to an evaluation of the capability of the production process of the monitored
product. The proposal for future research will be oriented toward the evaluation of this capability via
process capability indices that derive continuous data by using the classical method.

Keywords: Shewhart control charts; quality marks; variability control; production stabilization;
Pearson test

Citation: Malindzakova, M.;

Čulková, K.; Trpčevská, J. Shewhart 1. Introduction
Control Charts Implementation for
The production and processing of plastic moulding constitutes an important industry,
Quality and Production Management.
Processes 2023, 11, 1246. https://
as many traditional materials are replaced by cheap and inexpensive plastic materials.
doi.org/10.3390/pr11041246
Plastics are characterized by high specific strength, comparable thermal stability with
excellent chemical resistance, low cost, and their extremely light weight. Plastic materials
Academic Editor: Alina Pyka-Pajak
˛ are an integral part of the modern lifestyle, and their use has increased significantly over
Received: 22 February 2023 the past three decades. The total production of plastic materials in 2021 was 350.7 million
Revised: 4 April 2023 tons. European plastics production reached 57.2 Mt. PP, PVC, and HDPE are the major
Accepted: 13 April 2023 contributors to the total production of plastics in the world. Due to its versatile application
Published: 18 April 2023 and advantages, the consumption of plastic materials is increasing day by day. In 2021,
packaging, and building and construction applications were the two largest global plastics
markets (www.plasticseurope.org, accessed on 2 January 2023) [1].
This material also has highly beneficial properties in terms of shaping possibilities. By
Copyright: © 2023 by the authors. using suitable procedures, it is possible to achieve a huge variability of shapes. In order
Licensee MDPI, Basel, Switzerland. to maintain consistent quality level, it is possible to apply the Shewhart control chart to
This article is an open access article ensure the smooth running of the moulding production process.
distributed under the terms and
By definition, statistical process control (SPC) is a highly powerful tool used to reduce
conditions of the Creative Commons
the amount of variability to stabilise the process. Statistical process control problem
Attribution (CC BY) license (https://
solving can be implemented in any process. Additionally, SPC data can be used to predict
creativecommons.org/licenses/by/
future process performance. If the process is stable but operates outside of the required
4.0/).

Processes 2023, 11, 1246. https://doi.org/10.3390/pr11041246 https://www.mdpi.com/journal/processes

Processes 2023, 11, 1246 2 of 17

specification limits, the causes for the failures can be identified, removed, and the process
thus improved [2–7]. Processes will often operate in the in-control state for relatively
long periods. However, even if the process is relatively stable for a certain period of time,
no process can remain stable forever. Due to some determinable causes, every process
will eventually move to an unbalanced state with output parameters outside of the limit
requirements. The role of an SPC tool is to specifically identify such causes before their
full effect occurs. Subsequent process analysis and corrective measures can prevent non-
conforming process outputs from being produced [7].
From this toolset, the control chart is the most sophisticated approach. The control
charts are named after Walter A. Shewhart and are a popular tool implemented within
industrial processes, particularly when more than one variable needs to be controlled [3].
A great advantage of the control chart is it allows for the identification of change points in
time series and excess variability of a process. This functionality, along with its applicability
in real-time conditions, makes this SPC method highly suitable as an early failure warning
prevention system [5].
The popularity of control charts is primarily attributed to their functionality, as they
provide a reliable approach to the increase of productivity and offer diagnostic data in order
to evaluate the capability of a process. This can be achieved by implementing pc technology
that monitors and analyses these data in real time. Therefore, the careful implementation of
control charts can improve the performance of a process by identifying possible failures [3].
They present a graphical method by which a comparison is made based on the number of
samples that represent the investigated system. This is done in order to establish limits that
consider the typical process variability [4].
In practice, two main types of control charts are used, namely variables control charts
and attributes control charts. Each one includes two categories, based on whether the
standard value is specified or not. The selection of the relevant category should be based
on the quality attributes, on the specification of controlled parameters, and on the quality
requirements of the given product. Other specifications for the creation of control charts
include the sample size of the data provided, the number of sampling units, the specification
of control limits, and the central line. Once control charts are constructed, it can be
determined whether the process is stable or not [8].
The research observed so far does not take into account the implementation of She-
whart’s control diagrams for the needs of the production of plastic mouldings. The optimal
setting of monitored parameters contributes to the realization of the continuous operation
of the production process and to ensuring customer satisfaction. This paper proposes
a methodological framework for improving the quality parameters of plastic mouldings.
The goal of this paper is to determine the possible implementation of the Shewhart control
chart in the practical conditions of an industrial company that had not statistically managed
the quality of their output, with an aim to establish possible deformations of the products
during the production process. From a strategic point of view, the vision of the company is
focused not only on its development, but above all on ensuring its success in the market.
The scientific contribution of the presented contribution lies in the constant improvement,
innovation and advancement of the company to a higher level both in terms of technology
and in terms of quality in order to ensure the requirements of customers in the market are
met. In connection with the specific requirements of customers for products, the number of
products that are “tailor-made” for each customer is growing significantly. The investigated
company is focused on the production of specific products and for this reason it is necessary
to control the stability of the production process.

2. Literature Review
Efficient online quality control monitoring using control charts is a strategic aspect of
eliminating or reworking scrap to meet demand at the time specified by the production
plan. Starting the online monitoring of a quality characteristic by means of a control
chart at the beginning of a short production run is often a challenging issue for quality
Processes 2023, 11, 1246 3 of 17

practitioners. In the study of Castagliola et al. (2013), adapted control charts that had
typically been implemented with success in long runs to increase the performance of the
variable sample size strategy, were investigated for a chart used in a short run [9]. The
statistical performance of the VSS t-chart was compared with that of the fixed-parameter
(FP) t-chart for scenarios representing both fixed and unknown shift size, with the latter
a frequent situation in short-run manufacturing environments.
In the following year several authors studied the area of control charts, such as
Celano et al. (2012), who examined the way in which SPC inspection cost optimization is
constrained by the configuration of the manufacturing and the inspection activities [10].
Their study aimed to evaluating the economic performance of the Shewhart t-chart ver-
sus the Shewhart chart with known parameters. The expected economic loss associated
with the implementation of the Shewhart t-chart is acceptable with respect to the ideal
condition—a control chart with known parameters and where the cost optimization is
achieved without a statistical constraint limiting the number of expected false alarms.
Additionally, Li and Pu (2012) made an evaluation and computation based on statistical
performance measures for the two-sided Shewhart, cumulative sum, and exponentially
weighted moving average control charts for short-run production [11]. Their study com-
pares the efficiency of the three charts and the corresponding Q charts in the cases of
known and unknown nominal values of process parameters. Tasias and Nenes (2012) pre-
sented a new statistical process control model for the economic optimization of a variable-
parameter control chart monitoring a process operation where two assignable causes may
occur, one affecting the mean and the other the variance of the process [12]. Their study
used an economic (or an economic/statistical) optimization criterion for the time to the
next sampling instance, the size of the next sample, as well as the control limits of the
inspection. That is, all design parameters of the control scheme were selected so as to
minimize the total expected quality-related costs.
Bersimis et al. (2007), searched the basic procedures for the implementation of multi-
variate statistical process control via control charting, while also reviewing multivariate
extensions for all kinds of univariate control charts, such as multivariate Shewhart-type
control charts, etc. [13]. In addition, the study reviewed unique procedures for the
construction of multivariate control charts, based on multivariate statistical techniques
and partial least squares.
Consequently, Zhou and Tsung (2011) developed a new multivariate SPC methodology
for monitoring location parameters [14] in order to formulate the charting statistic by
incorporating the exponentially weighted moving average control (EWMA) scheme. This
control chart possesses some other favourable features: it is fast to compute with a similar
computational effort to the MEWMA chart; it is easy to implement; and it is also very
efficient in detecting process shifts.
Nenes et al. (2014) investigated a Shewhart control chart in a process with finite
production horizon because, when the production horizon is finite, the statistical properties
of a control chart are known to be a function of the number of scheduled inspections [15].
The study orientates to a short run producing a finite batch. In 2014, Chowdhury et al.
(2014) also studied a distribution-free Shewhart-type chart according to the Cucconi statistic,
called the Shewhart–Cucconi (SC) chart [16]. Additionally, they proprosed a follow-up
diagnostic procedure useful to determine the type of shift that the process may have
undergone when the chart signals an out-of-control process. Li (2015) proposed a new
nonparametric multivariate phase-II control chart, not dependent on any tuning parameter
and considered to be a natural generalisation of the generalised likelihood ratio chart to the
nonparametric setting [17]. The study showed that the proposed control chart performs
well across a broad range of settings and compares favourably with existing nonparametric
multivariate control charts.
Qiu (2018) gives some perspectives as to the robustness of conventional SPC charts
and the strengths and limitations of various nonparametric SPC charts [18]. Mukherjee and
Marozzi (2021) found that most of the charts are not robust when the real process parame-
 Qiu (2018) gives some perspectives as to the robustness of conventional SPC charts
and the strengths and limitations of various nonparametric SPC charts [18]. Mukherjee
and Marozzi (2021) found that most of the charts are not robust when the real process
parameters are unknown [19]. During their study, they proposed Shewhart-type non-
Processes 2023, 11, 1246 parametric monitoring schemes based on specific distance metrics for the surveillance 4 of 17 of
multivariate and high-dimensional processes.
The literature does not provide sufficient studies of the use of control charts in areas
ters are
of plastic unknownTherefore,
moulding. [19]. Duringthetheir
goalstudy, they
of this proposed Shewhart-type
contribution nonparametric
is to use control diagrams to
monitoring schemes based on specific distance metrics for the surveillance
ensure the operation of the production process and to provide important information of multivariate
aboutandthehigh-dimensional
process. The direct processes.
use of Shewhart control charts will then contribute to the
The literature does not provide sufficient studies of the use of control charts in areas
optimization of economic aspects, as well as to increase the accuracy of the dimensions of
of plastic moulding. Therefore, the goal of this contribution is to use control diagrams
the moulding products. In the case of this research, there is a particular emphasis on the
to ensure the operation of the production process and to provide important information
widthabout
and length of theThe
the process. moulding.
direct use of Shewhart control charts will then contribute to the
The aim of the paper
optimization of economic is aspects,
to suggest the as
as well use
to of a Shewhart
increase control
the accuracy chart
of the for the statis-
dimensions of
tical management of production
the moulding products. quality.
In the case The
of this topic there
research, is important because
is a particular the literature
emphasis on the
does width and length
not provide of the moulding.
research in the area of moulding. This method seems to be the most
proper for such conditions. is to suggest the use of a Shewhart control chart for the statistical
The aim of the paper
management of production quality. The topic is important because the literature does not
provide research in the area of moulding. This method seems to be the most proper for
3. Materials and Methods
such conditions.
3.1. Plastic Injection
3. Materials and Methods
A variety of technologies are used to process plastics, depending on the technologi-
3.1. Plastic Injection
cal properties of the plastic used, as well as the shape and function of the final product.
A variety of technologies are used to process plastics, depending on the technological
At present, injection moulding is the most widely used method. It is a cyclic discontin-
properties of the plastic used, as well as the shape and function of the final product. At
uous process. The injection moulding process is suitable for all types of plastics. The
present, injection moulding is the most widely used method. It is a cyclic discontinuous
principle of injection
process. moulding
The injection moulding is process
that the prepared
is suitable formolten
all typesmaterial is The
of plastics. injected intoofthe
principle
cavityinjection
of the moulding
metal mould, which, after solidification, forms a finished product.
is that the prepared molten material is injected into the cavity of the This
technology can produce
metal mould, which, complex products forms
after solidification, with good dimensional
a finished product. tolerances and acan
This technology good
surface finish.complex
produce The disadvantage
products with of good
this technology
dimensionalistolerances
the highand
initial costsurface
a good of the finish.
technol-
Theof
ogy and disadvantage
the mouldsof(Figure
this technology
1). is the high initial cost of the technology and of the
moulds (Figure 1).

Closing the injection mould Mould cavity filling and repressure

Plasticizing Opening of the mould, ejection of the moulding

Figure 1. Plastic injection cycle [20].

Figure 1. Plastic injection cycle [20].
The actual injection process has well-defined steps. The initial phase is the removal of
the granules
The from the hopper—a
actual injection process has helix is placed into
well-defined the melting
steps. chamber
The initial phase by isthethe
action of
removal
pressure and by the supplied heat by which the granules are plasticized, i.e.,
of the granules from the hopper—a helix is placed into the melting chamber by the action by which they
becomeand
of pressure liquid.
by In
thethe next phase,
supplied theby
heat plastic
whichis input into the cavity
the granules of the mouldi.e.,
are plasticized, by moving
by which
the helix as a piston, where it cools. However, during cooling the volume decreases, which
they become liquid. In the next phase, the plastic is input into the cavity of the mould by
could cause dents and nicks. Through subsequent overpressure, achieved by pressing
moving the helix
another materialas during
a piston, where
cooling, it cools. However,
the formation during
of depressions cooling
and sink marks the volume de-
is prevented.
creases,
Thiswhich could
pressure cause
can be dents andpressure,
the maximum nicks. Through
or weaker.subsequent overpressure,
After the product achieved
part has partially
by pressing another material during cooling, the formation of depressions
cooled, the mould is opened, and the finished part is removed from the mould (Figure 1). and sink
The whole cycle repeats [20–23].
The data come from the verification series. The specified quality features are measured
for all products, namely the width and length of the product. A digital calliper measures
the dimensions (Figure 1).
 marks is prevented. This pressure can be the maximum pressure, or weaker. After the
product part has partially cooled, the mould is opened, and the finished part is removed
from the mould (Figure 1). The whole cycle repeats [20–23].
Processes 2023, 11, 1246 The data come from the verification series. The specified quality features are meas- 5 of 17
ured for all products, namely the width and length of the product. A digital calliper
measures the dimensions (Figure 1).
The quality benchmarks
The quality dictatedictate
benchmarks the variability tolerance
the variability for the
tolerance fordimensions
the dimensions of the
of the
product,product, specifically
specifically its length
its length and width.
and width. The were
The data data were obtained
obtained duringduring a verification
a verification
series, inseries,
which in the
which the widths
widths and lengths
and lengths of all
of all 30 30 produced
produced piecespieces
werewere measured
measured within
within
three production
three production days. days.
Shewhart Shewhart
controlcontrol
charts charts were to
were used used to evaluate
evaluate the results.
the results. Shewhart
Shewhart control
control charts
charts
were used, since they assess the ability of the process to meet the criteria
were used, since they assess the ability of the process to meet the criteria set by the cus- set by the customer,
namely compliance with the specified limits and variability around the desired value.
tomer, namely compliance with the specified limits and variability around the desired
Shewhart control charts are useful for identifying deviations that indicate non-compliance
value. Shewhart control charts are useful for identifying deviations that indicate
with quality limits. The proposed methodological framework is subsequently implemented
non-compliance with quality limits. The proposed methodological framework is subse-
in a real case study of a company focused on the field of automobile production. The
quently research
implemented
aims atinimprovement,
a real case study of a company
innovation, focused on
and advancement to athe fieldlevel,
higher of automo-
not only in
bile production. The research aims at improvement, innovation, and advancement
terms of technology but also in compliance with quality parameters in order to satisfy to a the
higher level, not only
demands in termson
of customers ofthe
technology but also
market. During theinresearch
compliance with quality
we processed param-
according to the
eters in process,
order toillustrated
satisfy theindemands
Figure 2. of customers on the market. During the research we
processed according to the process, illustrated in Figure 2.

Specification of Molding Identification of

input material production evaluated parameters
(plastic granules) (length, width)

Application of
methods for the Priorities and

statistical quality limitations

management

Selection of the group

for detail analysis

Verification of the Finding the area of

production stability deviations

Evaluation of the
deviations reasons
Confirmation of the Rejecting of the
production stability production stability

Figure 2.Figure 2. Methodical

Methodical frame
frame for for the parameter
the parameter identification
identification of theof the production
production stability.
stability.
For the correct application of Shewhart control charts it is necessary to verify three
Forassumptions
the correct application of Shewhart control charts it is necessary to verify three
normalities of data: constant mean, constant standard deviation, and data
assumptions normalities of data:
independence. The normalityconstant mean,
of the data constant
is tested standard
by the Pearson deviation, and data
test of normality and the
Kolmogorov test as well as by a rapid test for determining normality, namely the test of
normality for asymmetry or excess. The paper applies Pearson’s normality test to test the
data normality.
 independence. The normality of the data is tested by the Pearson test of normality and
the Kolmogorov test as well as by a rapid test for determining normality, namely the test
of normality for asymmetry or excess. The paper applies Pearson’s normality test to test
the data normality.

Processes 2023, 11, 1246 3.2. Specification of Control Diagrams 6 of 17

To verify the stability of the process, it is appropriate to apply Shewhart’s control

charts. Shewhart has introduced control charts that serve the needs of statistical process
3.2. Specification
control. The actual of control
Control Diagrams
diagram (see Figure 3) shows the monitored parameters in a
traditional
To verify the stability the
xy-chart, where x-axis
of the represents
process, the time sequence
it is appropriate to applyand the y-axis
Shewhart’s the
control
measured values for
charts. Shewhart hasthe monitored
introduced parameter
control charts(width andthe
that serve length
needsofof
the product).
statistical The
process
chart is continuously
control. updated
The actual control at regular
diagram intervals
(see Figure and thus
3) shows provides aparameters
the monitored continuousin
overview of the
a traditional currentwhere
xy-chart, situation of therepresents
the x-axis process. the
These control
time sequencecharts
andmeasure
the y-axisand
the
compare
measured thevalues
control.
for A
thecombination of a control
monitored parameter diagram
(width for theofarithmetic
and length mean
the product). and
The a
chart
is continuously
control diagram updated
for the atvariation
regular intervals and thus provides
range evaluates a continuous
the parameters of the overview
productof
the current situation of the process. These control charts measure and compare the control.
(moulding).
A combination of a control diagram for the arithmetic mean and a control diagram for the
variation range evaluates the parameters of the product (moulding).

Selection
characteristics

Selection number
Shewhartcontrol
Figure3.3.Shewhart
Figure controlchart.
chart.

3.3. Control Diagram for the Range of Variation and Control Diagram for the Arithmetic
3.3. Control Diagram for the Range of Variation and Control Diagram for the Arithmetic
Control diagram for the selection range of variation [24,25]
Control diagram for the selection range of variation [24,25]
Test criterion R𝑅j == x𝑥max.,j., −−x𝑥min.,j.,
Testcriterion (1)
(1)
Central
Centralline
line(CL),
(CL),lower
lowercontrol
controlline
line(LCL)
(LCL)and
andupper
uppercontrol
controlline
line(UCL)
(UCL)

11 k
𝐶𝐿 =
CL =𝑅R=
= 𝑘 ∙· ∑ 𝑅
R (2)
(2)
k j =1 i

LCL = 𝐷D3∙· R
𝐿𝐶𝐿 = 𝑅 (3)
(3)

𝑈𝐶𝐿 = 𝐷 ∙ 𝑅 (4)
UCL = D4 · R (4)
Control diagram for the selection arithmetic mean [23,24]
Control
Test diagram for the selection arithmetic mean [23,24]
criterion
Test criterion
1 n
x j = · ∑ xij (5)
n i =1
Central line (CL), lower control Line (LCL) and upper control line (UCL)

= 1 k
k j∑
CL = x = · xj (6)
=1

=
LCL = x − A2 · R (7)

=
UCL = x + A2 · R (8)
 tomer requirements, namely 168.5 mm with a maximum permissib
mm. The measured data from the = 𝑥̿ −day
𝐿𝐶𝐿first 𝐴 ∙ are
𝑅 shown in Figure 4.

𝑈𝐶𝐿 = 𝑥̿ + 𝐴 ∙ 𝑅
Processes 2023, 11, 1246 7 of 17

4. Results
In the investigated company, the measurement values for the samples were tak
4. Results
three In
days. A digital calliper
the investigated company, measures the dimensions
the measurement values for the and every
samples wereday 30forpieces o
taken
were
threeobtained in this
days. A digital manner.
calliper The the
measures width of the and
dimensions product
every was determined
day 30 pieces of databased o
were obtained in this manner. The width of the product was
tomer requirements, namely 168.5 mm with a maximum permissibledetermined based on customer
deviation o
requirements, namely 168.5 mm with a maximum permissible deviation of ±0.4 mm. The
mm. The measured data from the first day are shown in Figure 4.
measured data from the first day are shown in Figure 4.

Figure 4. Measured data for the width parameter, first day.

4.1. Evaluation of the Parameter “Width”

Figure 4 shows that the manufactured products from day one
tomer’s requirements. This is the first production day of the test (v
Figure 4. Measured data for the width parameter, first day.
which
Figure the parameters
4. Measured data for thewere
widthset.
parameter, first day.
4.1. Evaluation of the Parameter “Width”
The first phase
Figure 4 shows
is to verifyproductsthe normality of the process. Figu
4.1. Evaluation of thethat the manufactured
Parameter “Width” from day one do not meet the cus-
generated histogram
tomer’s requirements. This isand frequency
the first production dayfunction
of the test does not correspond
(verification) series, in
Figure
which 4 showswere
the parameters thatset.
the manufactured products from day one do not meet th
bution.
tomer’s
This finding
Therequirements.
first phase is toThis
wasthe
verifyisthe
also confirmed
first production
normality
by Pearson’s
day Figure
of the process. of the 5test
test of a good
(verification)
shows that the ser
distribution.
generated
which the histogram
parameters and were
frequency
set.function does not correspond to the normal distribution.
This finding was also confirmed by Pearson’s test of a good fit with the normal distribution.
The first phase is to verify the normality of the process. Figure 5 shows th
generated histogram and frequency function does not correspond to the normal
bution. This finding was also confirmed by Pearson’s test of a good fit with the n
distribution.

Figure 5. Histogram and frequency function for the width parameter, first day.
Figure 5. Histogram and frequency function for the width parameter, first d
As the data do not show a normal distribution, it is not possible to verify the stability.
Figure
This is5.because
Histogram and frequency
the measured valuesfunction
from the for
firstthe
daywidth parameter,
were affected first
by the day. of all
setting
Asparameters.
process the data do not show a normal distribution, it is not possibl
ity. As the data do not show a normal distribution, it is not possible to verify the
This is because the measured values from the first day were affec
ity. This is because the measured values from the first day were affected by the sett
allprocess
all process parameters.
parameters.
 limits below the target value. An incorrect setting of the parameters of the prod
Processes 2023, 11, x FOR PEER REVIEW
ment and the subsequent occurrence of a non-conforming product causes th
formities of products.
Processes 2023, 11, 1246 8 of 17
The measured values for the “width” parameter for day two are shown in Fig
The corresponding chart in terms of variability with respect to the set limits shows
betterThe
results than the graph for the “width” parameter compared with the values o
measured values for the “width” parameter for day two are shown in Figure 6.
one.
TheOn the other chart
corresponding hand,in there
terms ofis variability
a challenge withtorespect
centring a set
to the specific process,
limits shows with all v
much
being
betterwithin the set
results than limitsfor
the graph below the target
the “width” value.
parameter An incorrect
compared with thesetting
values onforday
the param
one. On the other hand, there is a challenge to centring a specific
of the production equipment causes the subsequent occurrence of a non-confoprocess, with all values
being within the set limits below the target value. An incorrect setting for the parameters of
product.
the production equipment causes the subsequent occurrence of a non-conforming product.

Figure 6. Measured data of the width parameter, 2nd day.

Figure 7 shows the verification of normality, which indicates that the m

ues in theFigure
series
Figure have adata
6. Measured normal distribution,
of the width parameter, secondeven
day. though the second class has
6. Measured data of the width parameter, second day.
Figure 7 shows the verification of normality, which indicates that the measured values
than the first class.
in the series have a normal distribution, though the second class has fewer values than the
Figure 7 shows the verification of normality, which indicates that the mea
first class.
values in the series have a normal distribution, though the second class has fewer v
than the first class.

Figure 7. Histogram and the frequency function for the width parameter, second day.
Figure
Figure Histogram and and
7.7.Histogram the frequency function for the
the frequency width parameter,
function for thesecond
widthday.parameter, 2nd d

Thus,
Thus, aahistogram
histogram presents
presents data
data with with distribution.
a normal a normal distribution.
Because the firstBecause
assump- the fir
sumption applies, namely that the data show a normal distribution, it iswith
tion applies, namely that the data show a normal distribution, it is possible to proceed possible t
the stability verification. Based on the results of the Pearson test as well as the method of
ceed with the stability verification. Based on the results of the Pearson test as well
obtaining the values, it is possible to apply a combination of Shewhart control diagrams for
method of obtaining
sliding ranges the values,
and individual values.it is possible to apply a combination of Shewhart c
diagrams
Figurefor sliding
8 shows ranges
that and individual
the process values.
is stable, though the value 12 is at the lower control
limit, which 8may
Figure be caused
shows that bythetheprocess
two identical valuesthough
is stable, in succession.
the value 12 is at the lower c
limit, which may be caused by the two identical values in succession.
 Processes 2023, 11, x FOR PEER REVIEW 9 of 1
Processes 2023, 11, x FOR PEER REVIEW
Processes 2023, 11, x FOR PEER REVIEW 9 of 18
Processes 2023, 11, 1246 9 of 17

Figure 8. Shewhart control chart for sliding margins for the width parameter, second day.
Figure 8. Shewhart control chart for sliding margins for the width parameter, second day.
Figure 8. Shewhart
The course of control
the chart
valuesfor sliding margins
inforFigure for the
9 also width parameter,
confirms that thesecond day. is stable, as do
process
Figure The8.course
Shewhart control
of the valueschart
in Figure sliding
9 alsomargins
confirms forthat
the the
width parameter,
process second
is stable, as doday.
both ofthe
both ofThe
thecharts.
charts. For the width parameter, the measured values from day are
three are in
course of For the width
the values parameter,
in Figure the measured
9 also confirms that thevalues
processfrom day as
is stable, three
do both in
Figure
of the 10,ininwhich
charts. which
For the an improvement
anwidth parameter, inmeasured
the process canobserved.
be observed. The measured value
Figure 10,
The course of improvement
the values ininthe
the
Figureprocess values
can
9 also be from
confirms daythat
three
Thethe are in
measuredFigure
process 10,
values
is stable,
for
for thewidth
in the
whichwidth parameter
an improvement
parameter were
in the inappropriate
process can in in previous
be the the previous
observed. The two days
measured invaluesin terms
for theof varia
both of the charts. Forwere
the inappropriate
width parameter, the measuredtwo days values termsfromof varia-
day three
bility
width
bility and
and centring.
parameter
centring. were
TheThe measured
inappropriate
measured values
in
values thefor for
dayday
previous two
three three
days
are are evenly
in terms
evenly ofdistributed
variability
distributed aboveandabove
and and
Figure
centring.10,Thein which
measured anvalues
improvement
for day in are
three theevenly
process can be observed.
distributed above and The measured
below the
below the target value, without a significant number of values in a row on one side. Itside.
below the target value, without a significant number of values in a row on one is It i
for the
target width
value,
possibletotoconsider
possible
parameter
without
considerthisathis were
significant inappropriate
number
a well-established
a well-established of values inin
process,
process,
the
a row
though
previous
on
though one two
side. Itdays
is
it is necessary
it is necessary
in
possibleterms
to verify
to verify its
of
it
bility
to and
consider
stabilityand
stability centring.
this a The measured
well-established
andcapability.
capability. values
process, thoughfor day
it is three
necessary are toevenly
verify distributed
its stability abov
and capability.
below the target value, without a significant number of values in a row on one sid
possible to consider this a well-established process, though it is necessary to ve
stability and capability.

Figure 9. Shewhart control chart individual values for the width parameter, second day.
Figure 9. Shewhart control chart individual values for the width parameter, second day.
Figure 9. Shewhart control chart individual values for the width parameter, second day.

Figure 9. Shewhart control chart individual values for the width parameter, second day.

Figure 10. Measured data for the width parameter, third day.
Figure 10. Measured data for the width parameter, third day.

Figure 10. Measured data for the width parameter, third day.
 Figure 10. Measured data, the width parameter, 3rd day.
Figure 11 shows that the measured values have a normal distributi
appropriate if there were more values in the last product class compare
Processes 2023, 11, 1246
Figureclasses,
vious 11 shows
forthat the measured
example values class.
in the middle have aThe
normal distribution.
bell-shaped It wo
division
10 of 17
propriate if there were
pronounced. Themore values inprocess
production the last product classorder,
is in good compared to middle
as the the previoc
symbolinclose
for example to the required
the middle value
class. The of 168.5 division
bell-shaped mm. would be more pronou
Figure 11 shows that the measured values have a normal distribution. It would
production processifisthere
be appropriate in awere
good more valuesasinthe
order, the last product
middle classhas
class compared
a classwith the
symbol clos
previous classes, for example in the middle class. The bell-shaped division would be more
quired pronounced.
value, namely 168.5 mm.
The production process is in good order, as the middle class has a class symbol
close to the required value of 168.5 mm.

Figure 11. Histogram and frequency function for the width parameter, third day.

From
Figure the Shewhart
Histogram
11.11. control
and frequency chart
function for (Figure
the width 12)third
parameter, it can
day. be determined th
Figure Histogram and frequency function for the parameter width, 3rd day.
stable even if it contains a series of five values in a row below the target
From the Shewhart control chart (Figure 12) it can be determined that the process is
not suitable.
stable The use
even if it contains of the
a series upper
of five valuescontrol limitthe
in a row below except for the
target value, first
which value is
is not
suitable. The use of the upper control limit except for the first value is very good.

Figure 12. Shewhart control chart for sliding margins, the width parameter, 3rd day

Figure 12. Shewhart control chart for sliding margins for the width parameter, third day.
Figure 12. Shewhart control chart for sliding margins for the width parameter, thi
The second pair of constructed charts are charts (Figure 13) for individual values.
Additionally, all values are found in the chart and so this process can be labelled, from
The second
the perspective of the pair of constructed
calculated regulatory limitscharts are charts
of Shewhart’s (Figure
control charts, 13) for ind
as stable.
Additionally,
A maximum of half allofvalues are found
the regulatory in the and
limits is reached, chart and
points so regularly
move this process
around can b
the central line.
the perspective of the calculated regulatory limits of Shewhart’s control c
A maximum of half of the regulatory limits is reached, and points move re
the central line.
 Processes 2023, 11, x FOR PEER REVIEW 11 of

Processes 2023, 11, x FOR PEER REVIEW

Processes 2023, 11, 1246 11 of 17

Figure 13. Shewhart control chart for individual values for the width parameter, third day.

4.2. Evaluation of the Parameter “Length”

Figure 13. Shewhart control chart for individual values for the width parameter, third day.
In the same way as the evaluation for the “width” parameter, the measured valu
Figure 13. Shewhart control chart for individual values for the width parameter, third d
for the
4.2. “length”
Evaluation parameter
of the Parameter were also available. The measurements were performed ov
“Length”
three Indays, and way
the same thirty
as values were collected.
the evaluation Theparameter,
for the “width” customer’s therequest
measured for the length
values for of t
4.2.
the
Evaluation
“length”
of the were
parameter
Parameter
also
“Length”
available. The measurements were performed over three
product was 344.4 mm with a maximum permissible deviation of ±0.45 mm. Figure
days,In
presents andthe
the same
thirty
measuredway
values were ascollected.
the for
values evaluation
The for the
the customer’s
“length” “width”
request for the
parameter parameter,
length
for the measur
of the product
all manufactured comp
was 344.4 mm with a maximum permissible deviation of ±0.45 mm. Figure 14 presents
for the
nents “length”
that parameter
are required to meetwere also available.
the customer specifiedThe measurements were perfor
limits.
the measured values for the “length” parameter for all manufactured components that are
three days,
required andthethirty
to meet values
customer were
specified collected. The customer’s request for the len
limits.
product was 344.4 mm with a maximum permissible deviation of ±0.45 mm.
presents the measured values for the “length” parameter for all manufacture
nents that are required to meet the customer specified limits.

Figure 14. Measured data for the length parameter, first day [20].
Figure 14. Measured data for the length parameter, first day [20].
To verify the normal distribution, it is appropriate to apply Pearson’s chi-square
goodness of fit test (Figure 15). Because the values of the fifth class have a class symbol
To verify the normal distribution, it is appropriate to apply Pearson’s chi-squa
lower than the target value, i.e., the generated histogram has a shape—a ridge shape— that
goodness of measurement
can indicate fit test (Figure 15). Because the values of the fifth class have a class symb
errors.
lowerFigure
than the targetthat
16 shows value, i.e., the generated
the variability histogram
of measurements haslow
is very a shape—a
concerningridge
the shape
that can indicate
customer measurement
requirements. For processerrors.
purposes, an improvement can be seen in both the
Figure 14.and
centring Measured
the value data for the
variability, length parameter,
indicating firstisday
that the process [20].slightly above the
centred
target value.
To verify the normal distribution, it is appropriate to apply Pearson’s c
goodness of fit test (Figure 15). Because the values of the fifth class have a cla
lower than the target value, i.e., the generated histogram has a shape—a ridg
that can indicate measurement errors.
 Processes 2023, 11, x FOR PEER REVIEW 12 of 18

Processes 2023, 11, x FOR PEER REVIEW

Processes 2023, 11, 1246 12 of 17

Figure 15. Histogram and frequency function for the length parameter, first day [20].

Figure 16 shows that the variability of measurements is very low concern

customer requirements. For process purposes, an improvement can be seen in
centring and the value variability, indicating that the process is centred slightly ab
target
Figure 15. value.
Histogram and frequency function for the length parameter, first day [20].

Figure 16 shows that the variability of measurements is very low concerning the
customer requirements. For process purposes, an improvement can be seen in both the
centring and the value variability, indicating that the process is centred slightly above the
target value.
Figure 15. Histogram and frequency function for the length parameter, first day [20].
Figure 15. Histogram and frequency function for the length parameter, first day [20].

Figure 16 shows that the variability of measurements is very low concern

customer requirements. For process purposes, an improvement can be seen in b
centring and the value variability, indicating that the process is centred slightly ab
target value.

Figure 16. Measured data for the length parameter, second day [20].
Figure 16. Measured data for the length parameter, second day [20].
Figure 16.
InMeasured
the firstdata for the
class, lengthare
there parameter, second day
components [20]. a
with size of approximately 344.4 m
theInIn the first class,
following
the first five
five there are components
classes
class,classes
there are
represent
components
with
a
with
a size component,
longer of approximately
a size of approximately
344.4 can
which mm,also
and be see
the following represent a longer component, which can also344.4 mm,in
be seen and
the
histogram.
the According
followingAccording
five classes to the ahistogram
represent (Figurewhich
17), poor centring is confirmed,
histogram. to the histogramlonger component,
(Figure 17), poor centring can also be seen
is confirmed, in the
though the
thehas
seta normal
histogram.
set has a normal
According distribution.
to the
distribution. histogram (Figure 17), poor centring is confirmed, though
the set has a normal distribution.

Figure 16. Measured data for the length parameter, second day [20].

In the first class, there are components with a size of approximately 344.4 m
the following five classes represent a longer component, which can also be seen
histogram. According to the histogram (Figure 17), poor centring is confirmed,
the set has a normal distribution.

Figure 17. Histogram and the frequency function for the length parameter, second day.
Figure 17. Histogram and the frequency function for the length parameter, second day.
Figure 17. Histogram and the frequency function for the length parameter, second day.
After a positive finding of a normal distribution, it is necessary to verify the stability
of the process. The constructed control chart for the sliding ranges (Figure 18) does not
indicate any problem, though point 16 is at the lower control limit.
 Processes 2023, 11, x FOR PEER REVIEW 13 of 18

After a positive finding of a normal distribution, it is necessary to verify the stability

of the process.
After The constructed
a positive finding of a control
normal chart for the sliding
distribution, ranges to
it is necessary (Figure
verify 18)
the does not
stability
indicate any problem, though point 16 is at the lower control limit.
of the process. The constructed control chart for the sliding ranges (Figure 18) does not
Processes 2023, 11, 1246 13 of 17
indicate any problem, though point 16 is at the lower control limit.

Figure 18. Shewhart control chart for sliding margins for the length parameter, second day.
Figure 18. Shewhart control chart for sliding margins for the length parameter, second day.
Figure 18. Shewhart control chart for sliding margins for the length parameter, second day.
The above Shewhart control chart for individual values (Figures 18 and 19) show
The aboveare
that all Shewhart
within control chartlimits,
for individual
it canvalues (Figures
that 18
theand 19) show that
Thepoints
above Shewhart the control
control chart for andindividual be stated
values (Figures process
18 and 19) is statis-
show
all points
tically are within the control limits, and it can be stated that the process is statistically
that allmastered.
points areThe evaluation
within for the
the control length
limits, andparameter
it can be was also
stated performed
that on the
the process third
is statis-
mastered.
day. As Theofevaluation
part the for of
centring thethe
length parameter
process, it is was also
possible toperformed
see an on the thirdover
improvement day. the
As
tically
part of
mastered.
the centring
The
of
evaluation
the process,
for
it
the
is
lengthto
possible
parameter
see an
was also performed
improvement over the
on the third
previous day.
previous
day. day.ofSignificant
As part the centringchanges in
of theare variability
process, it is are not visible
possible to see and this was confirmed
an improvement over theby
Significant
the changes
calculation of instandard
the variability not visible
deviation. and this was confirmed by the calculation
previous day. Significant
of the standard deviation.changes in variability are not visible and this was confirmed by
the calculation of the standard deviation.

Figure 19.
Figure Shewhart control
19. Shewhart control chart
chart for
for individual
individual values
values for
for the
the length
length parameter,
parameter, second
second day.
day.
Processes 2023, 11, x FOR PEER REVIEW
FigureFollowing
19. Shewhart
thecontrol chart for individual
determination of the values for the length parameter,
the normal secondverify
day.
Following the determination of normal distribution,
distribution, itit is
is possible
possible to
to verify the
the
stability of
stability of the
the process
process of
of the
the respective
respective production batch
batch (Figure 20).
Following the determination of theproduction (Figure
normal distribution, it is20).
possible to verify the
stability of the process of the respective production batch (Figure 20).

Figure 20. Histogram and the frequency function for the length parameter, third day.
Figure 20. Histogram and the frequency function for the length parameter, third day.

Figure 21 presents the Shewhart control chart for sliding margins, which in
the mastered process. All values oscillate around the target value and are wit
 Figure20.
Figure 20.Histogram
Histogram and
and thethe frequency
frequency function
function forlength
for the the length parameter,
parameter, third day.
third day.
Processes 2023, 11, 1246 14 of 17
Figure2121presents
Figure presents thethe Shewhart
Shewhart control
control chartchart for sliding
for sliding margins,
margins, which which in
indicates
the mastered
the masteredprocess.
process.AllAll values
values oscillate
oscillate around
around the target
the target value value
and are and are wit
within the
Figure
regulatory 21 presents
limits. The the Shewhart
exception iscontrol
point chart
nine, for sliding
which is margins,
located which
on the
regulatory limits. The exception is point nine, which is located on the lower contr indicates
lower the
control limit.
mastered process.
This All values oscillate around the target value it and are within
thatthe regulatory compo-
This does
doesnot
notindicate
indicate a significant
a significantproblem, rather
problem, rather indicates
it indicates successive
that successive
limits.
nents The exception is point nine, which is located on the lower control limit. This does
nentshave
havethe
thesame
same dimension.
dimension.
not indicate a significant problem, rather it indicates that successive components have the
same dimension.

Figure 21. Shewhart control chart for sliding margins for the length parameter, third day.
Figure 21. Shewhart control chart for sliding margins for the length parameter, third day.
Figure
The21.Shewhart
Shewhartcontrol
controlchart
chartforforindividual
sliding margins
valuesfor the length
(Figure 22) canparameter,
be used third day.
to describe
The Shewhart control chart for individual values (Figure 22) can be used
the way in which all values of the test criteria are within the control limits. The only to describe
TheinisShewhart
the way
problem which control
all values
the grouping chart
ofofthe testfor
thethree individual
criteria
values are values
within
located onthe
one (Figure
control
side of 22)
the can
limits. Thebe used
only
central to d
line. This
problem is the grouping
the way isinacceptable
situation of
which allbut the
values three values
of the test
is not suitable located on one
for acriteria side of
are managed
statistically the
within thecentral line. This
control limits. T
process.
situation is acceptable but is not suitable for a statistically managed process.
problem is the grouping of the three values located on one side of the central lin
situation is acceptable but is not suitable for a statistically managed process.

Figure 22. Shewhart control chart for individual values, parameter length, 3rd day.

5. Discussion and Conclusions

The investigated product is a license plate holder. As it is a supply element for the
automotive industry, high demands are placed on its quality. However, according to Ercole
et al. (2021), the results of the production of plastic material can be supported by the
production of succinic acid by acid succinogenes immobilized on plastic materials (50%
polypropylene) [26]. The results of the presented study may be helpful as there are still
limitations to the current technologies for recycling and processing waste plastics [27],
resulting in significant environmental problems. Other research has shown the potential
for solvent-based processing to produce secondary plastic materials from E-waste for cross-
industry reuse [28]. The using of plastics can also be applied to the manufacture of utility
objects to help cater to the immediately emerging need for the rapid production of personal
protective equipment [29].
Processes 2023, 11, 1246 15 of 17

The data used in this paper come from the production of the verification series, but the
investigated company subsequently switched to serial production for this product series.
In the verification series, three production batches were produced within three days and
within one month. The production management identified two critical parameters: the
width and length of the product. By implementing Shewhart control diagrams, it was
possible to diagnose adverse effects and then take corrective and preventive measures. The
purpose of Shewhart’s control diagrams is to obtain information about interventions in the
process. Using Shewhart control diagrams, it is possible to achieve and maintain the pro-
duction process at the required level to ensure the required product parameter compliance.
Shewhart control charts indicate problems in the parameter setting of the production pro-
cess. It is crucial to monitor the variability and maintain the centre values of the respective
measurements when setting up the production process. Within the investigated case study
scenario, at the beginning of the verification series for parameters “width” and “length”,
the findings show that both set parameters did not meet the customer’s requirements. In
both cases, measurement errors were evident on the first day of the verification series.
This study also confirms that, from the point of view of statistical calculations on the first
day of the verification series, the production process needs to be set to specific required
parameters. The following measurements show an improvement on the second and third
verification days.
Statistical analyses show that the set parameters still need to be met in the first two
days of the test series. An improvement in the process is observed on the third day of
the test series. Based on the setting and fine-tuning of the parameters in the case of
the third day of the test series, it can be concluded that the process is well set, which is
confirmed by verifying the stability of the process. An essential condition for setting the
control charts is to observe the correct chronological arrangement and regular acquisition
of measured values.
Due to inaccuracy and mistakes during the measurements, repeated measurements
were necessary. The purpose of managing the smooth running of the production process is
to regulate, coordinate and control the production process through management tools and
statistical quality control. The current turbulent and dynamically developing environment
requires, in addition to compliance with delivery dates, the use of ecologically appropriate
materials and technologies and compliance with product quality parameters. On the other
hand, it is a complex, laborious and demanding tool; collecting the necessary data and
evaluating them objectively and effectively is necessary. The implementation of statistical
regulation in the management and control of the production process allows the company’s
top management to eliminate the consumption of used material and used energy and
reduce the total production costs.
Solving tasks in the future must be oriented toward the evaluation of the capability
of the production process of the monitored product. The proposal for the following
research will be oriented to evaluating the capability through process capability indices
for continuous data using the classical method for Cp, Cpk, Pp, and Ppk and with a
set minimum limit of 1.33. Another suggestion for process capability assessment is the
calculation of Cpmk and Cpm indices.
This specific and realized research has demonstrated, in the form of a case study of
the conditions of a real company, that it can offer that company an advantage due to the
quality control of the “width” and “length” parameters. The priority of the organization
in the long term is toward an effort to comply with the required parameters “width” and
“length” determining the quality of manufactured products. In the case of the increasing
variability, this result may have a negative impact on the basic parameters of the product.
The implementation of Shewhart’s control diagrams contributes to the elimination or
removal of the deformation of plastic mouldings, which in turn leads to ensuring the
correct functioning of their related production process. The current modern era requires an
increase in the quality parameters of individual production processes, which will ultimately
have an impact on the competitiveness of the company and an increase in market share.
Processes 2023, 11, 1246 16 of 17

Shewhart control charts are useful for identifying deviations that indicate non-compliance
with quality limits. The proposed methodological framework was subsequently imple-
mented in a real case study of a company focused on the field of automobile production.
The research showed the improvement, innovation and advancement to a higher level, not
only in terms of technology but also in compliance with quality parameters in order to
satisfy the demands of customers on the market.
Realized Shewhart control diagrams present the measurements taken during test
batches while setting up the machines and establishing the process itself. The scientific
contribution consists in monitoring and investigating the deformation of plastic mould-
ings with a connection to the method of statistical quality control. The positive scientific
contribution of the application of Shewhart’s control diagrams derives from the way it
leads to improved quality and higher efficiency. The everyday reality of the company
centres around ensuring the continuous operation of the production process, minimizing
non-conforming products and eliminating downtime. Applying these three motivations
helps to ensure quality, reduce production line times and eliminate waste.
Ultimately, the company must respond flexibly to change and it is in the company’s
interest to prosper and increase its market share. It is the application of Shewhart control
diagrams that contributes to the elimination of non-conforming products as well as to the
elimination of downtime. Through the application of Shewhart control charts, production
of non-conforming products was reduced by 10%. It should be emphasized that the result
must have a positive effect on the basic parameters of the product. The current modern era
requires the streamlining of production processes by applying Shewhart’s control charts,
the application of which has begun to represent a major trend in manufacturing companies.
The implementation of Shewhart control charts is aimed at increasing productivity as well
as eliminating costs incurred.
The calculation of process capability through process capability indices, namely Cp,
Cpk, Cpm, Cpmk and Pp, Ppk will be the subject of further studies. The use of more
modern and reliable indices will focus on both variability and on the degree to which the
optimal value of the observed quality is achieved in order to achieve further improvement
of the results.

Author Contributions: Conceptualization, M.M. and K.Č.; methodology, M.M., J.T. and K.Č.; soft-
ware, M.M.; validation, K.Č., M.M. and J.T.; formal analysis, K.Č.; investigation, J.T.; resources, M.M.;
data curation, K.Č. and J.T.; writing—original draft preparation, K.Č.; writing—review and editing,
M.M.; visualization, M.M.; supervision, K.Č., J.T. and M.M.; project administration, K.Č.; funding
acquisition, M.M. All authors have read and agreed to the published version of the manuscript.
Funding: This work is supported by the Scientific Grant Agency of the Ministry of Education, Science,
Research, and Sport of the Slovak Republic and the Slovak Academy Sciences as part of the research
project VEGA 1/0408/23.
Data Availability Statement: Data is unavailable due to privacy.
Conflicts of Interest: The authors declare no conflict of interest.

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