*Corr. Author’s Address: Jacob of Paradies University, Teatralna 25, 66-400 Gorzów Wielkopolski, Poland, aperec@ajp.edu.pl 367
Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 367-375 Received for review: 2023-05-17
© 2023 The Authors. CC BY 4.0 Int. Licensee: SV-JME Received revised form: 2023-07-31
DOI:10.5545/sv-jme.2023.647 Original Scientific Paper Accepted for publication: 2023-08-21
Optimization of Abrasive Waterjet Cutting  
by Using the CODAS Method  
with Regard to Interdependent Processing Parameters
Perec, A. – Kawecka, E. – Radomska-Zalas, A. – Pude, F.
Andrzej Perec
1,*
 – Elżbieta Kawecka
1
 – Aleksandra Radomska-Zalas
1
 – Frank Pude
2,3
1 
Jacob of Paradies University, Faculty of Technology, Poland 
2 
Steinbeis Consulting Center High-Pressure Waterjet Technology, Germany 
3 
Inspire AG, ETH Zurich, Switzerland
The paper shows performance optimization effects of steel machining by abrasive water jet (AWJ). An innovative combinative distance-based 
assessment method (CODAS) is implemented for the optimization of cutting parameters like pump pressure, feed rate, and abrasive flow 
rate over cutting depth, and cut kerf surface roughness. The CODAS algorithm is among those based on measuring the distance between a 
scenario (in this case processing parameters in terms of performance and quality indicators) - and a certain benchmark. A benchmark is a 
specific hypothetical set of processing parameters devised or determined from available data. To determine the best set of process control 
parameters, a CODAS approach was performed with some weighting determinations. To set the initial parameters of the weights, it was 
proposed to calculate based on entropy weight method (EWM), that measures output value dispersion in cutting process. This technique 
simplifies multiple compound responses by preserving a single response.
Keywords: abrasive waterjet cutting, process optimization, CODAS method, maximum cutting depth, minimum surface roughness
welding, copper, metal matrix composite, boron carbide
Highlights
• 	 Effect of jet pressure, traverse speed, and mass flow rate of on chromium-nickel-molybdenum steel cut surface roughness was 
introduced. 
• 	 An optimal set of control parameters to reach the highest cutting depth and smallest surface roughness of cut kerf was 
determined. 
• 	 A significant abasement in the cost of the experiments by diminution the number of required tests and shortening the time to 
perform with high precision of results was reached.
0  INTRODUCTION
Optimization of control parameters is used wherever 
many control parameters significantly affect the result. 
Examples include advanced industrial processes like 
epoxidation [1], polymerization [2], and advanced 
manufacturing technologies [3] to [5]. 
The cutting with an abrasive waterjet (AWJ) is 
one of the imported methods classified as advanced 
manufacturing technology. It is used in many 
industries, including aerospace [6] and [7] automotive 
[8], manufacturing [9] and [10] and even in medicine 
[11] to [13]. However, a poorly designed process can 
be costly and time-consuming and optimizing it can 
ensure that it is as efficient, quality [14] and [15] and 
effective as possible.
Optimizing the control parameters of AWJ 
machining process is essential for achieving the 
desired cutting results with maximum efficiency, 
quality, and minimum waste. The control parameters 
optimization of the AWJ machining process can 
be achieved by using statistical methods such as 
design of experiments (DOE), especially Taguchi 
method [16], response surface methodology (RSM) 
[4], artificial intelligence techniques such as neural 
networks and genetic algorithms, expert systems, and 
approximate metaheuristic methods. These methods 
can help identify the optimal combination of control 
parameters that maximize the desired cutting results, 
reduce waste, and improve efficiency.
There are also methods from the field of decision 
support or multi-criteria decision-making (MCDM) 
that can be successfully used to optimize the AWJ 
process [17] and [18].
The CODAS method is a valuable tool for solving 
MCDM problems. It allows to consider multiple 
criteria simultaneously and to balance the trade-offs 
between them. The method can be used in fields such 
as mechanical engineering, among others.
The entropy-CODAS method belongs to a 
multi-criteria decision-making technique used for 
optimization of chosen problems. It is based on the 
concept of entropy, which is a measure of uncertainty 
or disorder. The basic idea behind the method is to 
minimize the overall distance between the alternatives 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 367-375
368 Perec, A. – Kawecka, E. – Radomska-Zalas, A. – Pude, F.
and the ideal solution while maximizing the diversity 
among the alternatives.
An innovative entropy-CODAS method is 
implemented for the optimization of cutting depth, 
cut surface roughness, and angle of cut kerf was 
conducted. 
The dimension and distribution of the used 
abrasive grains have a noteworthy influence on 
the efficiency of the cutting process by AWJ. In the 
cutting head takes place the intensive disintegration of 
abrasive materials during the creation of the abrasive 
jet. The disintegration of chosen abrasive materials 
grains was tested after forming in the cutting head [19] 
and [20]. Additionally, it allows to carry out recovery 
analysis for the recycling possibility.
Hlavacova et al. [21] introduced the study of 
common quality steels cut by AWJ and observed the 
relations concerning the mechanical characteristics 
of quenched steels and the chosen surface roughness 
parameters. The differentiation of the steel 
microstructure was the essential property for the 
cutting quality because the higher the difference 
in the hardness of the structural constituents in the 
inhomogeneous microstructure was, the higher were 
the surface roughness values after cutting.
Perec and Musial [22] conducted research on the 
use of one of the methods based on decision support, 
namely the VIKOR method for optimizing the 
parameters of wear-resistant structural steel cutting by 
AWJ. 
However, Perec et al. [23] modeled and optimized 
the AWJ cutting process of tool steel based on the 
RSM.
Other possibility of hard material machining 
presented Kumari and Acherjee [24]. Authors 
concentrated on using criteria importance through 
inter-criteria correlation (CRITIC) and CODAS 
multicriteria decision-making methods to assess 
the performance of proposed approach while 
selecting the best advanced machining process for 
machining titanium from the eight most often used 
as AWJ machining, ultrasonic machining, chemical 
machining, electron beam machining, laser beam 
machining, electrochemical machining, electro 
discharge machining, and plasma arc machining. 
Material removal rate, shape feature, work material 
type, tolerance and surface finish, power requirement, 
and cost were the criteria used to evaluate and pick the 
best advanced manufacturing process.
Sivalingam et al. [25] investigated the effect 
of cutting process parameters on Inconel 718 alloy 
turning in dry and (atomized spray cutting fluid) 
ASCF cutting environments. The cutting parameters 
were adjusted using desirability functional analysis, 
and two types of MCDM methods were investigated: 
additive ratio assessment method (ARAS) and 
CODAS. Both MCDM approaches yielded identical 
results in the form of minimal surface roughness, 
machining cost, power consumption and maximizing 
tool life, compared with dry machining.
Al-Tamimi and Sanjay [26] presented an 
intelligent machining model which used contemporary 
techniques, based on CODAS and several other as 
artificial neural network (ANN), adaptive neuro-fuzzy 
inference systems, and particle swarm optimization 
(ANFIS-PSO) approach for minimizing resulting 
force, specific cutting energy, and maximizing metal 
removal rate in superalloys machining.
Malaga et al. [27] presented study tended to 
identify the proper material for metal additive 
manufacturing, using MCDM approach. Information 
entropy method (IEM) and CODAS were taken 
to establish the priority order of materials. The 
meaningful material properties were used as the 
material criterium for the analysis. The decision-
making techniques were deployed using real data of 
materials.
Sivalingam et al. [28] also presented the CODAS 
multi-criteria decision-making techniques and additive 
ratio assessment method for predicting the internal 
combustion engine radiator performance under 
27 different operating conditions using multiwall 
carbon nanotubes based nanofluid. The outcomes of 
the regression analysis designated those substantial 
input factors for enhancing thermal transfer with this 
radiator.
Due to the difficulties in the milling of steel and 
the difficulty in the proper selection of cutting tools, 
cutting conditions and parameters of the cutting 
process, Abas et al. [29] performed a multi-response 
optimization using a CODAS method in combination 
with criteria importance through inter-criteria 
correlation (CRITIC) with satisfactory results.
The CODAS method can be also used for 
support of optimal selection for example supplier 
selection [30], wind energy plant location selection 
[31], dam construction material selection [32], and for 
sustainable material selection in construction projects 
with incomplete weight information [33].
Krajcarz and Spadlo [34] published experimental 
research of the geometric accuracy of cylindrical holes 
made by a high-pressure jet of water. The tests were 
conducted according to a three-level Box-Behnken 
design. Changes in the input parameters during high-
pressure abrasive water jet cutting resulted in the 
occurrence of geometric inaccuracies. The values of 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 367-375
369 Optimization of Abrasive Waterjet Cutting by Using the CODAS Method with Regard to Interdependent Processing Parameters  
the correlation coefficient confirmed that the greatest 
influence of the cylindrical holes was cutting speed.
The state of art includes assorted studies on the 
optimization of cutting parameters, including cutting 
depth, cut surface roughness, and angle of cut kerf, 
using different methods such as entropy-CODAS, 
VIKOR, RSM, and multicriteria decision-making 
techniques like CODAS, ARAS, and desirability 
functional analysis. These studies focused on cutting 
types of materials, such as common quality steels, 
wear-resistant structural steel, tool steel, Inconel 
718 alloy, and superalloys. Additionally, it can be 
observed that the CODAS method was also applied to 
other fields. 
However, to date, the CODAS method has not 
been used in the optimization of AWJ machining, 
which defines a research gap and an area for 
potentially new research.
The objective of this paper is to utilize entropy-
CODAS to gain an optimal combination of control 
parameters for maximum cutting depth and minimum 
surface roughness and to uncover the individual result 
of each control parameter on cutting depth, width of 
the cut kerf and it surface roughness.
1  MATERIALS
1.1  Abrasive Materials
As abrasive material the crushed rock garnet type 
J80A from Jinhong Mining located in Jiangsu, China 
was used. A sample view of grain shape and grain size 
distribution is presented in Fig. 1.
From the details of the mineral content shown 
in Table 1, more than 90 % of this type of garnet is 
Almandine. 
Almandine belongs to the silicate mineral group 
as part of the larger garnet group, which includes 
several other types of minerals with similar crystal 
structures. It has the chemical formula Fe
3
Al
2
(SiO
4
)
3
, 
which shows it contains both iron (Fe), aluminum 
(Al), and silicon (Si) atoms.
Almandine is typically found in metamorphic 
rocks such as mica schists, gneisses, and amphibolites. 
It is usually red to reddish-brown in color, although 
it can also appear purple or black. Almandine is a 
hard mineral with a Mohs hardness over 7.5, making 
it suitable for use as an abrasive material. Other 
properties are shown in Table 2.
a)     b) 
c) 
Fig. 1.  Garnet abrasive grains: a) SEM view; b) optical microscope 
view; and c) grain size distribution
Table 1.  Garnet chemical properties
Chemical composition [%]
Fe
2
O
3
SiO
2
TiO
2
Al
2
O
3
FeO CaO MgO MnO
17 39 0.05 21 8 9.5 5 0.4
Mineral content [%]
Amandine Ilmenite Omphacite Rutile Quartz Hornblende Silica
90-96 1.0 1.5 0.6 <0.1 <0.5 <0.5
Table 2.  Garnet physical properties
Size Unit Value
Density [kg/dm
3
] 3.8-4.1
Bulk gravity [kg/dm
3
] 2.3-2.4
Mohs hardness 7.5-8.0
Conductivity [S/m] <25
Acid solubility (HCL) [%] <1.0
Grain shape Sub angular
In addition to its use as an abrasive, almandine is 
also used as a gemstone because of its deep red color 
and durability.
1.2  Cut Material
As target material 18CrNiMo7-6 steel for medium 
to high core strength engineering applications up to 
62 HRC when carburized, hardened, and tempered 
was chosen to be cut. In this steel chromium-nickel-
molybdenum were used as strengthening agents 
(Table 3). 

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 367-375
370 Perec, A. – Kawecka, E. – Radomska-Zalas, A. – Pude, F.
Table 3.  18CrNiMo7-6 steel chemical composition [34]
[%] C Si Mn P S Cr Ni Mo Cu
min 0.15 0.15 0.5 - - 1.5 1.4 0.25 -
max 0.21 0.4 0.9 0.02 0.02 1.8 1.7 0.35 0.4
It is a high hardenability, high toughness case-
hardening steel, generally supplied in the annealed 
condition.
It can also be used in uncarburized form as a 
high tensile steel, which when suitably hardened and 
tempered can be utilized for various applications 
requiring good tensile strength and toughness.
Despite difficult to cut is used extensively by all 
industry sectors for components and shafts requiring 
high surface wear resistance, high core strength 
and impact properties. The strength properties are 
presented in Table 4.
Table 4.  18CrNiMo7-6 steel typical mechanical properties [35] and 
[36] 
Youngs 
module 
[GPa]
Poisson’s 
ratio
Shear 
module 
[GPa]
Density 
[kg/m
3
]
Tensile 
strength 
[MPa]
Yield  
strength 
[MPa]
210 0.3 80 7800 700 520
2  EXPERIMENTAL 
2.1  Test Rig and Test Method
The cutting tests were carried out on the WaterJet 
CNC OMAX 60120 machining center (Fig. 2). 
Fig. 2.  AWJ cutting process: 1) target material, 2) focusing tube,  
3) AWJ, 4) cutting head, 5) abrasive inlet, and 6) cutting head cover
The materials were cut by perpendicular to the 
workpiece directed AWJ, and a linear moving with a 
specific traverse speed. The thickness of the samples 
was selected to prevent complete separation of 
material and an accurate determination of the cutting 
depth accordingly. 
The process of AWJ 18CrNiMo7-6 steel cutting 
was conducted using the following parameters:
• pressure: 360 MPa; 380 MPa; 400 MPa,
• traverse speed: 50 mm/min; 150 mm/min and 
250 mm/min,
• the abrasive flow rate: 250 g/min; 350 g/min and 
450 g/min,
• abrasive material; garnet #80 (from crushed 
rock), 
• water nozzle ID: 0.33 mm,
• focusing tube ID: 0.76 mm,
• stand-off distance: 2 mm.
2.1.1  Cut Kerf Geometry 
The effect of the AWJ on the material is a cut kerf. Its 
details are shown in Fig. 3a and the actual view in Fig. 
3b. The depth of the cut groove is denoted as h
c
, its 
width as W
t 
, at top and W
b 
at bottom, and the angle of 
kerf inclination as δ.
a)     b) 
Fig. 3.  Cut kerf dimensions: a) schematic, and  
b) optical microscope view
2.1.2  Surface Roughness
For roughness measurement the Sku (kurtosis) was 
chosen. This parameter expands the profile (line 
roughness) parameter Rku three dimensionally. Sku, is 
used to evaluate sharpness in the height distribution 
[37]. It is calculated from the following equation:
 Sku
Sq A
Zx yd xdy
A
 









11
4
4
,. (1)
This parameter concerns the height distribution 
and is suitable for evaluating the abrasion, when (Fig. 
4):
•	 Sku = 3: normal distribution, 
•	 Sku > 3: height distribution is sharp, and
•	 Sku < 3: height distribution is even.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 367-375
371 Optimization of Abrasive Waterjet Cutting by Using the CODAS Method with Regard to Interdependent Processing Parameters  
Fig. 4.  Sku roughness parameter details
Roughness measurements were made on the high-
definition Olympus DSX1000 optical microscope. 
The measurement area was set as 953 	 mm × 953 mm. 
Its location is shown in Fig. 4. The measurement 
signal was filtered with the Gaussian filter.
Fig. 5.  Cut kerf roughness measurement location
2.2  CODAS Method
Combinative distance-based assessment (CODAS) 
is a multi-criteria decision-making method that 
was introduced in a paper by Ghorabaeeet al. [38]. 
CODAS algorithm belongs to the class of those based 
on measuring the distance between a scenario (in our 
case, it will be the processing parameters in terms of 
performance and quality indicators), and a certain 
benchmark.
A benchmark is a certain hypothetical set of 
processing parameters, imagined or determined from 
available data. The idea behind the CODAS method is 
as follows: we are looking for a worst-case scenario, a 
negative ideal. We check how far each scenario (each 
set of machining parameters) is distanced from this 
worst-case scenario in the Euclidean sense. The farer 
away a set of parameters is from the counter-ideal, the 
better it is (and vice-versa).
In CODAS, we are interested in the negative 
ideal. First measure of this method is the distance of 
the scenario from this negative ideal is checked here, 
and this distance is calculated using the Euclidean 
metric. It is this metric that we consider the most 
intuitive: the square root of the sum of the squares 
of the differences of the values of the corresponding 
coordinates.
The secondary measure is the taxicab distance 
which is related to the indifference space. The 
taxicab distance equation is grounded on the concept 
that the length between two points is determined by 
following a grid, rather than following a straight line. 
The equation is the sum of the absolute value of the 
difference of x values and the absolute value of the 
difference of y values.
The steps of the proposed CODAS method are 
presented as follows:
Step 1. Construct the decision-making matrix 
(X), shown as follows:
 X  



















x
xx
xx
x
x
xx x
ij
nm
m
m
nn nm
11 12
21 22
1
2
12
 
 

, (2)
where x
ij
 (x
ij
 ≥ 0) denotes the performance value of 
i
th
 alternative on j
th
 criterion (i ∈ {1, 2, …, n} and  
j ∈ {1, 2, …, m}).
Step 2. Calculate the normalized decision matrix. 
We use linear normalization of performance values as 
follows: 
 n
x
x
jN
x
x
jN
ij
ij
i
ij
b
i
ij
ij
c










max
min
,
if
if
 (3)
where N
b
 and N
c 
represent the sets of benefit and cost 
(non-beneficial) criteria, respectively.
Step 3. Calculate the weighted normalized 
decision matrix. The weighted normalized 
performance values are calculated as follows:
 r
ij
	= 	w
j
	· 	n
ij
	, (4)
where w
j
	 (0 	 < 	w
j
	 < 	 1) denotes the weight of j
th
 
criterion, and 
 
j
m
j
w



1
1. (5)
To establish the entropy factor (e
i,j
) exploiting the 
projection value of the alternative, the equation looks 
as follow:
 e
m
TT
ij
i
n
ij ij ,, ,
ln
ln , 


1
1
 (6)

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 367-375
372 Perec, A. – Kawecka, E. – Radomska-Zalas, A. – Pude, F.
and the entropy weight of the j
th
 index is determined 
by equation:
 w
e
e
i
ij
i
n
ij






1
1
1
,
,
. (7)
This entropy technique was used to determine 
the level of individual weights. In this technique, 
the number of choices, and different criteria get to 
appraise multiple criteria optimizations on basis 
establishing a comparative decision matrix. If the 
number of choices (mass flow rate, pressure, and feed 
rate) getting as ‘M’, and the numbers of conditions 
are cutting depth surface rough-ness and angle of cut 
kerf get as ‘N’ then relative decision matrix having a 
dimension of M×N. 
Step 4. Determine the negative-ideal solution as 
follows:
 ns ns
j
m
 



 1
, (8)
 ns r
j
i
ij
= min. (9)
Step 5. Calculate the Euclidean and taxicab 
distances of alternatives from the negative-ideal 
solution, shown as follows:
 Er ns
i
j
m
ij j




1
2
, (10)
 Tr ns
i
j
m
ij j



1
. (11)
Step 6. Construct the relative assessment matrix, 
shown as follows:
 Ra h
ik
m

 n
, (12)
 hE EE ET T
ik ik ik ik
    
 , (13)
where k ∈ {1, 2, …, n} and 𝜓 denotes a threshold 
function to recognize the equality of the Euclidean 
distances of two alternatives, and is defined as 
follows:
 


x
x
x
 







1
0
if
if
, (14)
where τ is the threshold parameter that can be set by 
decisionmaker. It is suggested to set this parameter 
at a value between 0.01 and 0.05. If the difference 
between Euclidean distances of two alternatives is less 
than τ, these two alternatives are also compared by the 
taxicab distance. In this study for the calculations was 
used τ = 0.02.
Step 7. Calculate and rank the alternatives 
according to the decreasing values of assessment 
score ( Η
i
):
 Hh
i
k
n
ik



1
. (15)
The alternative with the highest Η
i
 factor is the 
best choice among the alternatives.
3  RESULTS AND DISCUSSION
The results shown in Table 5, while Table 6 displays 
the calculation effects of the normalizing, weighted 
normalized performance values, Euclidean and 
taxicab distances of alternatives, assessment score 
factor and their ranks.
Table 5.  Cutting process tests results
No AFR p Vp hc Sku
1 250 360 50 7.48 2.57
2 250 380 150 5.09 3.01
3 250 400 250 2.93 2.63
4 350 360 150 4.99 3.72
5 350 380 250 3.06 2.65
6 350 400 50 8.70 2.46
7 450 360 250 3.31 2.66
8 450 380 50 7.59 3.83
9 450 400 150 4.89 2.65
Table 6.  CODAS coefficients and rank
No Ri(b) Ri(nb) Ei Ti H Rank
1 0.55 0.35 0.35 0.36 1.23 2
2 0.37 0.29 0.17 0.31 -0.42 5
3 0.22 0.34 0.11 0.35 -1.20 9
4 0.37 0.24 0.15 0.25 -0.58 6
5 0.23 0.33 0.10 0.35 -1.02 8
6 0.64 0.36 0.44 0.38 2.03 1
7 0.24 0.33 0.11 0.35 -1.00 7
8 0.56 0.23 0.34 0.25 1.14 3
9 0.36 0.33 0.18 0.35 -0.36 4
The calculated H
i
 represents the better the status, 
the higher values it takes. Out of all H
i
 value in the 
frame of the reference sequence is the best combination 
of parameters and is thereby recommended. 
For these tests, the recommended values for 
control parameters (highlighted row in Table 6) are as 
follows:
• abrasive feed rate:  350 g/min,
• pressure:  400 MPa,
• traverse speed:  50 mm/min.

Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 367-375
373 Optimization of Abrasive Waterjet Cutting by Using the CODAS Method with Regard to Interdependent Processing Parameters  
Examples of the effects of machining with 
the control parameters optimally determined by 
this method are shown in Fig. 5. Numerous traces 
of erosion of the material by abrasive grains were 
observed here. They become visible in the form 
of parallel machining footprints. They are visible 
especially in Fig. 6a in the form of parallel lines 
located on macrograins, at an acute angle. There is 
no chaotic arrangement of traces on adjacent grains, 
which indicates good cutting conditions.
a) 
b) 
Fig. 6.  Cut kerf surface at optimal conditions:  
a) SEM view, and b) optical microscope view
4  CONCLUSIONS
The conducted research confirmed the equity of 
applying the method in multi-criteria optimization 
of the 18CrNiMo7-6 steel cutting process by AWJ. 
The CODAS method transforms the multiple 
characteristics of cutting process into the individual 
H
i
 coefficient, which significantly simplifies the 
computation. The CODAS method determines the 
ranks of evident from computational results by 
optimal machining variable combination.
Optimal condition from cutting depth and 
roughness surface was achieved at following control 
parameters:
• abrasive feed rate: 350 g/min,
• pressure: 400 MPa,
• traverse speed: 50 mm/min.
Future studies will be conducted on the impact of 
other control parameters.
5  NOMENCLATURES
X		 decision-making matrix,
AFR abrasive flow rate, [g/min]
p	 	 pressure, [MPa]
Vp  traverse speed, [mm/min] 
hc	 	 depth of cut, [mm] 
N
b
  set of benefit criteria,
N
c
  set of cost (non-beneficial) criteria,
n
ij
  normalized decision matrix,
ns	 	 negative solution,
Sku	 	 surface roughness factor (curtosis), [ mm] 
R
a
		 relative assessment matrix value,
r
ij
		 weighted normalized performance value,
Ei	 	 Euclidean distances of alternatives,
Ti 	 	 taxicab distances of alternatives,
H
i
  assessment score factor, 
ψ	 	 threshold function.
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Strojniški vestnik - Journal of Mechanical Engineering 69(2023)9-10, 367-375
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