Revista de Ciencias Tecnológicas (RECIT). Volumen 3 (1): 10-22
Revista de Ciencias Tecnológicas (RECIT). Universidad Autónoma de Baja California ISSN 2594-1925
Volumen 9 (3): e498. Julio-Septiembre, 2026. https://doi.org/10.37636/recit.v9n3e498
1 ISSN: 2594-1925
Research article
Development of a machining allowance model for rectangular
pocket milling operations
Desarrollo de un modelo de tolerancia de mecanizado para operaciones de fresado
de cavidades rectangulares
Tran Thanh Tung1 , Nguyen Thi Anh2, Nguyen Xuan Quynh3, Tran Vu Minh3,*
1Faculty of Engineering Mechanics and Automation, VNU University of Engineering and
Technology, Hanoi, Vietnam
2Faculty of Mechanical Engineering, Thuyloi University, Hanoi, Vietnam
3School of Mechanical Engineering, Ha Noi University of Science and Technology, Hanoi, Vietnam
*Corresponding author: Tran Vu Minh, School of Mechanical Engineering, Ha Noi University of Science and
Technology, Hanoi, Vietnam. E-mail: minh.tranvu@hust.edu.vn ORCID: 0000-0003-0621-9217.
Received: June 12, 2026 Accepted: June 23, 2026 Published: July 8, 2026
Abstract.- Machining allowance plays a critical role in determining productivity, energy consumption, cutting force, and
dimensional accuracy in pocket milling operations, yet it is commonly selected empirically in industrial practice. This
study proposes an analytical, optimization-based framework for determining an efficient machining allowance for
rectangular pocket milling. Mathematical models are developed to quantify total energy consumption, machining time, and
peripheral cutting force as functions of cutting parameters and allowance distribution between rough and finish milling
stages. A multi-objective optimization problem is formulated and solved using the Non-dominated Sorting Genetic
Algorithm II (NSGA-II) to simultaneously minimize energy consumption, machining time, and cutting force. Experimental
validation is conducted on polymer workpieces and compared with CAM-recommended parameters. The results show that
the balanced Pareto-optimal solution reduces machining time from 5 min 58.52 s to 2 min 26.47 s, corresponding to an
approximate 59% reduction, and decreases the peripheral cutting force from 698.18 N to 382.73 N, corresponding to an
approximate 45% reduction, while achieving the target pocket depth of approximately 22 mm and eliminating the ~2 mm
over-machining observed under CAM-default settings. A minimum-energy solution further reduces energy consumption
from 0.44 kWh to 0.267 kWh, but at the cost of a significantly increased cutting force (1203.45 N). These results indicate
that treating machining allowance as an optimization variable enables systematic trade-offs among productivity, energy
consumption, cutting force, and dimensional accuracy, offering a practical alternative to conventional CAM-based
parameter selection.
Keywords: Machining allowance; Pocket milling; Multi-objective optimization; Energy consumption; Cutting force.
Resumen. - El margen de mecanizado desempeña un papel fundamental en la determinación de la productividad, el
consumo de energía, la fuerza de corte y la precisión dimensional en las operaciones de fresado de cavidades; sin embargo,
en la práctica industrial, suele seleccionarse de forma empírica. Este estudio propone un marco analítico y de optimización
para determinar un margen de mecanizado eficiente en el fresado de cavidades rectangulares. Se desarrollan modelos
matemáticos para cuantificar el consumo total de energía, el tiempo de mecanizado y la fuerza de corte periférica en función
de los parámetros de corte y la distribución del margen entre las etapas de fresado de desbaste y acabado. Se formula y
resuelve un problema de optimización multiobjetivo mediante el algoritmo genético de clasificación no dominada II
(NSGA-II) para minimizar simultáneamente el consumo de energía, el tiempo de mecanizado y la fuerza de corte. Se
realiza una validación experimental en piezas de polímero y se compara con los parámetros recomendados por CAM. Los
resultados muestran que la solución óptima de Pareto equilibrada reduce el tiempo de mecanizado de 5 min 58,52 s a 2 min
26,47 s (lo que corresponde a una reducción aproximada del 59 %) y disminuye la fuerza de corte periférica de 698,18 N
a 382,73 N (lo que corresponde a una reducción aproximada del 45 %), al tiempo que alcanza la profundidad de cavidad
objetivo de aproximadamente 22 mm y elimina el sobremecanizado de ~2 mm observado con la configuración
predeterminada de CAM. Una solución de energía mínima reduce aún más el consumo de energía de 0,44 kWh a 0,267
kWh, pero a costa de un aumento significativo de la fuerza de corte (1203,45 N). Estos resultados indican que tratar el
margen de mecanizado como una variable de optimización permite realizar compensaciones sistemáticas entre
productividad, consumo de energía, fuerza de corte y precisión dimensional, ofreciendo una alternativa práctica a la
selección de parámetros convencional basada en CAM.
Palabras clave: Margen de mecanizado; Fresado de cavidades; Optimización multiobjetivo; Consumo de energía; Fuerza
de corte.
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1. Introduction
Machining operations are commonly performed through a sequence of roughing and finishing stages
in order to balance productivity and dimensional accuracy [1-5]. During rough machining, large
amounts of material are removed at high material removal rates, often at the expense of geometric
accuracy and surface integrity. To ensure that the final component meets design specifications, a
certain amount of material, referred to as machining allowance or excess material, is intentionally left
on the workpiece after rough machining and subsequently removed during the finishing stage.
Machining allowance plays a critical role in determining the overall performance of the machining
process [6-9]. As schematically illustrated in Figure 1, machining allowance represents the
intermediate material layer between the rough-machined surface and the final finished surface. This
allowance must be sufficient to compensate for surface irregularities, elastic deformation of the tool–
workpiece–fixture system, tool runout, and positioning errors generated during rough machining. At
the same time, excessive allowance during finishing leads to increased cutting forces, accelerated tool
wear, higher energy consumption, and increased vibration, which may deteriorate surface quality and
reduce machining stability.
Figure 1. Machining allowance removal in milling during the finishing stage.
Pocket milling is one of the most fundamental and widely used machining operations in modern
manufacturing, particularly in the production of molds, dies, structural components, and mechanical
parts with internal cavities [10-12]. Owing to its flexibility and high material removal capability, CNC
pocket milling is extensively applied in aerospace, automotive, and general mechanical industries.
However, despite its apparent simplicity, achieving both high productivity and stable machining
quality remains challenging. The pocket milling process involves complex interactions among cutting
parameters, tool geometry, material properties, and machine tool dynamics, which become especially
pronounced during multi-stage machining operations.
To illustrate the geometric characteristics of pocket milling and the relationship between rough and
finish machining stages, a schematic of a typical rectangular cavity milling process is shown in Figure
2. The figure highlights the finishing surface and the remaining machining allowance after rough
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milling, which plays a critical role in determining cutting stability, surface integrity, and dimensional
accuracy during the finishing stage.
Figure 2. Rectangular cavity milling and definition of machining allowance.
In industrial practice, machining allowance is often determined empirically based on operator
experience or general machining guidelines. However, such experience-based selection does not
account for the complex and nonlinear relationships between allowance, cutting parameters, and
machining performance. An improper allocation of machining allowance can lead to several
undesirable consequences. Excessive remaining allowance increases the cutting load during the
finishing stage, resulting in elevated cutting forces, accelerated tool wear, higher energy consumption,
and increased machine vibration. Conversely, insufficient allowance after rough milling limits the
ability of the finishing process to correct geometric errors and surface irregularities, thereby preventing
the achievement of the required dimensional accuracy and surface roughness. These issues are
particularly critical in the milling of pockets with sharp corners or rectangular geometries, where tool
engagement conditions vary significantly along the toolpath.
Previous studies on milling have primarily focused on cutting force modeling, toolpath optimization,
chatter stability, and surface quality prediction [13-20]. While these studies have contributed
significantly to the understanding of milling mechanics, machining allowance is often treated as a fixed
or predefined parameter rather than a decision variable to be optimized. Only limited research has
attempted to establish a quantitative relationship between machining allowance, machining time, and
quality constraints, especially for rectangular pocket geometries commonly encountered in practical
manufacturing. As a result, process planners still lack a systematic and scientific method for
determining an efficient machining allowance that balances productivity, energy efficiency, cutting
load, and quality requirements.
To address this research gap, this study proposes a mathematical and optimization-based model for
determining an efficient machining allowance in rectangular pocket milling operations, with particular
emphasis on rectangular pockets. The proposed model establishes analytical relationships between
machining allowance, toolpath characteristics, cutting parameters, machining time, energy
consumption, cutting force, and dimensional accuracy. Based on these relationships, a multi-objective
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optimization framework is formulated to determine the optimal allocation of machining allowance
between roughing and finishing stages. The objectives are to minimize total machining time while
satisfying constraints related to surface roughness and dimensional accuracy.
2. Materials and methods
2.1. Energy, Machining Time, and Cutting Force Models
The analytical formulations adopted in this study are based on established energy, time, and cutting
force models reported in previous machining research [21-27]. These models are reformulated here in
a compact form to support multi-objective optimization of machining allowance in rectangular pocket
milling.
The total energy consumption during the pocket milling process is expressed as the sum of the energy
associated with feed motions and material removal,
(1)
where
denotes the equivalent power consumed during non-cutting feed motions, including idle,
spindle rotation, and feed drive power, and
is the cutting power required for chip formation. The
variables and represent the feed time and cutting time, respectively. The cutting power is modeled
as a function of the primary cutting parameters using a generalized empirical relationship,
(2)
where is the spindle speed,
is the feed rate, and are the axial and radial depths of cut,
and and are material- and machine-dependent coefficients.
The total machining time is decomposed into feed time and cutting time,
(3)
For pocket milling, the feed time is estimated from the total tool travel length and the feed rate,
(4)
while the cutting time is determined from the ratio of the removed material volume to the material
removal rate (MRR),
(5)
where is the cutting speed and is the tool diameter. Through these relations, the machining
allowance directly influences both feed and cutting times by governing the number of passes and the
total material volume removed.
The characteristic peripheral cutting force is incorporated using a compact analytical expression
suitable for optimization [5, 14],
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(6)
where is the tangential cutting force coefficient, is the effective radial depth of cut, and is
the average uncut chip thickness, which depends on the feed rate and machining allowance. This
formulation allows cutting force to be evaluated efficiently without repeating detailed force
derivations.
Based on the above models, the pocket milling process is formulated as a multi-objective optimization
problem aiming to simultaneously minimize total energy consumption, total machining time, and
characteristic cutting force,
The decision variable vector is defined as
where represents the machining allowance distribution between rough and finish stages, is the
radial depth of cut, is the spindle speed, and
and
are the feed rates in rough and finish
milling, respectively. Practical machine-tool and process constraints are imposed to ensure feasibility.
The resulting non-convex multi-objective optimization problem is solved using the NSGA-II algorithm
to obtain a set of Pareto-optimal solutions.
2.2. Optimization Model for Pocket Milling
This section presents a simulation-based optimization framework for a two-stage pocket milling
process consisting of rough and finish machining. The proposed model aims to simultaneously
optimize three mutually conflicting performance objectives: total energy consumption , total
machining time , and total characteristic peripheral cutting force .
These objectives exhibit pronounced trade-off relationships. In general, reducing machining time leads
to increased energy consumption and higher cutting forces, whereas minimizing cutting forces requires
conservative cutting parameters that significantly prolong machining time. Similarly, reducing energy
consumption often necessitates smaller cutting steps, which lowers productivity. Consequently, the
problem is formulated as a non-convex multi-objective optimization problem that cannot be effectively
solved using classical gradient-based or linear programming methods. To address this challenge, the
Non-dominated Sorting Genetic Algorithm II (NSGA-II) is adopted to identify Pareto-optimal
solutions [28-32].
The optimization problem involves five decision variables: the machining allowance distribution
parameter between rough and finish stages , the effective radial depth of cut , the spindle
speed , the feed rate in rough milling
, and the feed rate in finish milling
. The
bounds of these variables are defined according to machine tool capabilities and tool limitations. To
reflect practical operating conditions of the GSM-800 CNC milling machine, the spindle speed and
feed rates are discretized to multiples of 10 rpm and 10 mm·min-1, respectively.
In this study, the NSGA-II algorithm was implemented using a population size of 100 individuals and
200 generations. Tournament selection was used to select parent solutions, while simulated binary
crossover and polynomial mutation were applied to generate offspring. The crossover probability and
mutation probability were set to 0.90 and 0.10, respectively. The optimization process was terminated
when the maximum number of generations was reached. These parameters were selected to maintain
sufficient solution diversity and convergence stability while keeping the computational cost reasonable
for the present machining optimization problem.
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To ensure feasibility and industrial relevance, several technical constraints are imposed. The ratio
between the feed rates of the rough and finish milling stages is constrained to maintain machining
stability,
(7)
ensuring that rough milling is performed at a higher feed rate without inducing excessive tool loading.
In addition, the feed rate is constrained by the spindle speed through the kinematic relationship.
(8)
where is the spindle speed, is the number of cutter teeth, and is the feed per tooth. For a 10-mm
end mill with four cutting edges machining polymer material, stable cutting conditions are
experimentally observed when
(9)
These constraints effectively reduce vibration, prevent tool overload, and eliminate infeasible solutions
during the optimization process.
The first optimization objective is total energy consumption. As coolant is not required for polymer
machining, the total energy is expressed as the sum of idle, spindle rotation, feed drive, and cutting
energies and is computed as
(10)
The second objective is total machining time, defined as the sum of feed time and cutting time, both
of which depend on pocket geometry, machining allowance distribution, feed rates, spindle speed, and
cutting depths.
The third objective is the characteristic peripheral cutting force. For optimization purposes, the cutting
force is incorporated using a compact analytical expression,
(11)
where is the tangential cutting force coefficient, is the effective radial depth of cut, and is
the average uncut chip thickness, which depends on the feed rate and machining allowance. This
formulation enables the cutting force to be directly evaluated within the optimization framework
without repeating the detailed force derivation. Among the decision variables, the radial depth of cut
and feed rate have the strongest influence on
and therefore dominate the force-related trade-offs in
the optimization process.
The decision variables are encoded as a real-valued vector
At each
generation, NSGA-II applies non-dominated sorting and crowding-distance evaluation to preserve
solution diversity while promoting convergence toward the Pareto-optimal front. To ensure
comparability among objectives with different magnitudes, all objective functions are normalized
using min–max normalization,
(12)
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The final optimization problem is formulated to minimize the normalized energy consumption,
machining time, and cutting force subject to the defined technical constraints.
After obtaining the Pareto-optimal front, three representative solutions were selected for experimental
comparison: the CAM-recommended baseline, the balanced Pareto-optimal solution, and the
minimum-energy Pareto solution. The balanced Pareto-optimal solution was selected using the
minimum normalized Euclidean distance to the ideal point. In this method, the ideal point corresponds
to the individual minimum values of the normalized energy consumption, machining time, and cutting
force. Therefore, the selected balanced solution represents a compromise among the three objectives,
rather than the optimum of a single objective only.
2.3. Experimental setup for pocket milling
Figure 3. Experimental setup for pocket milling on a polymer workpiece using a CNC milling machine.
To validate the proposed energy–time–force models and the multi-objective optimization framework,
a series of milling experiments were conducted on a CNC milling machine under controlled conditions.
The experiments were designed to replicate the two-stage pocket milling process (rough and finish
milling) assumed in the analytical and optimization models.
The experimental setup consisted of a three-axis CNC milling machine equipped with a vertical
spindle, as shown in Figure 3. A polymer workpiece was rigidly clamped on the machine table using
a mechanical fixture to ensure stable positioning during machining. A flat end mill with a diameter of
10 mm was employed for all experiments. The tool geometry and cutting parameters were selected in
accordance with the ranges defined in the optimization model to ensure consistency between
simulation and experimental validation.
Pocket milling was performed in two consecutive stages. In the first stage, rough milling was applied
to remove the majority of the material using relatively higher feed rates and larger radial depths of cut.
In the second stage, finish milling was carried out with reduced feed rates and cutting widths to
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improve dimensional accuracy and surface quality. The machining allowance distribution between the
two stages followed the values defined by the decision variable .
Machining time was recorded from the CNC machining cycle for each pocket. Energy consumption
was measured from the electrical energy used during the corresponding machining cycle, including
spindle rotation, feed motion, and cutting. The peripheral cutting force was obtained using the force
measurement system during the milling tests, and the characteristic maximum force value was used
for comparison among the different machining strategies. After machining, the pocket depth was
measured at representative positions using a depth gauge. The surface condition was evaluated through
visual inspection of tool marks, wall uniformity, and bottom-surface quality. These measurement
procedures provided a consistent basis for comparing the CAM-recommended baseline, the balanced
Pareto-optimal solution, and the minimum-energy Pareto solution.
3. Results and discussion
At the beginning of the experimental validation, the pocket milling strategies corresponding to the
baseline and Pareto-optimal solutions were implemented in a commercial CAM environment. Figure
4 illustrates the machining setup and toolpath generation in Autodesk Fusion, where three rectangular
pockets were programmed on the same polymer workpiece using different parameter sets. Each pocket
corresponds to a distinct machining strategy—namely the CAM-recommended baseline, the balanced
Pareto-optimal solution, and the minimum-energy Pareto solution—allowing a direct and fair
comparison under identical geometric and fixturing conditions. The visualization confirms consistent
tool orientation, pocket geometry, and machining sequence, thereby ensuring that the observed
differences in machining time, energy consumption, cutting force, and surface quality arise solely from
the selected cutting parameters and machining allowance distributions.
Figure 4. CAM implementation of rectangular pocket milling strategies in CAD software.
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The proposed optimization framework was experimentally evaluated through three rectangular pocket
milling trials conducted on polymer workpieces (workpiece size: 300 × 300 × 50 mm; pocket
geometry: 50 × 30 × 20 mm). The three trials correspond to (i) a conventional baseline strategy using
CAM-recommended parameters, (ii) a Pareto-optimal solution representing a balanced trade-off
among energy, time, and cutting force, and (iii) an extreme Pareto solution targeting minimum energy
consumption. Representative cavity geometries obtained under these three conditions are shown in
Figure 5,6,7.
Figure 5. CAM-recommended baseline (smooth surface, long time, over-cut).
Using the cutting parameters automatically recommended by Autodesk Fusion, the first cavity (Figure
5) was machined at a spindle speed of 2000 rpm with a step-over of 6 mm and a programmed depth of
−20 mm. Under these conditions, the machining time was 5 min 58.52 s, the measured peripheral
cutting force was approximately 698.18 N, and the energy consumption was about 0.44 kWh. This
strategy produced the smoothest surface finish among the three cases, as evidenced by the visually
uniform cavity walls and bottom in Figure 5. However, the measured pocket depth reached
approximately 24 mm, indicating an over-machining error of nearly 2 mm relative to the target depth.
This depth deviation may be attributed to the CAM-default allowance setting, accumulated depth-
control error during multi-pass machining, and the difference between the programmed toolpath depth
and the actual material removal obtained on the polymer workpiece. Although the conservative CAM-
recommended parameters ensured good surface quality and stable cutting behavior, the excessive
machining time and significant dimensional deviation limit their suitability for precision pocket
milling applications.
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Figure 6. Balanced Pareto-optimal solution (shortest time, lowest force, poor surface).
The second cavity (Figure 6) was machined using a parameter set selected from the Pareto front as the
most balanced solution, with a spindle speed of 5000 rpm, a reduced step-over of 4.7301 mm, and a
programmed depth of −17.9927 mm. This configuration reduced the machining time to 2 min 26.47 s,
corresponding to approximately 41% of the baseline machining time. At the same time, the peripheral
cutting force decreased to 382.73 N, which is about 55% of that observed in the CAM-recommended
case, while energy consumption increased slightly to 0.52 kWh. As shown in Figure 6, visible tool
marks appear on the cavity surface, indicating degraded surface quality compared with the baseline.
Nevertheless, the measured pocket depth was approximately 22 mm, closely matching the desired
geometry and eliminating the over-cut observed in the first case. This result demonstrates that the
optimized machining allowance distribution effectively improves dimensional accuracy and
productivity, albeit at the expense of surface finish.
Figure 7. Minimum-energy Pareto-optimal solution.
The third cavity (Figure 7) corresponds to the Pareto solution that minimizes energy consumption.
This cavity was machined at 5000 rpm with a large step-over of 9.9945 mm and a programmed depth
of −17.996 mm. The measured energy consumption was the lowest among all trials, approximately
0.267 kWh, confirming the effectiveness of the energy-oriented optimization objective. However, the
machining time remained relatively long at 5 min 47.48 s, only marginally shorter than the CAM-
recommended case, while the peripheral cutting force increased substantially to approximately
1203.45 N, representing nearly 1.7 times the baseline force and more than three times that of the
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balanced solution. As illustrated in Figure 10(c), the cavity exhibits a relatively smooth overall
appearance with less pronounced tool-path marks than in Figure 10(b), but the bottom surface remains
uneven. Although no chatter or tool jamming was observed, the high cutting force implies increased
mechanical loading, which may negatively affect tool life and machine reliability in long-term
operation.
A comparative analysis of the three cavities shown in Figure 5-7 clearly reveals the trade-off structure
predicted by the multi-objective optimization model. The CAM-recommended strategy (Figure 5)
prioritizes surface quality and conservative cutting but results in long machining time and unacceptable
depth deviation. The minimum-energy strategy (Figure 7) achieves the lowest energy consumption but
introduces excessive cutting force with only marginal improvement in machining time. In contrast, the
balanced Pareto-optimal solution (Figure 6) provides the most practically relevant compromise,
achieving significant reductions in machining time and cutting force while maintaining acceptable
dimensional accuracy. These results confirm that the proposed optimization framework enables
informed selection of machining parameters based on specific production priorities, rather than relying
solely on default CAM-generated settings.
4. Conclusion
This study presented an analytical and multi-objective optimization framework for determining an
efficient machining allowance in rectangular pocket milling. By explicitly incorporating machining
allowance into energy, time, and cutting force models, the proposed approach enables a quantitative
evaluation of trade-offs in two-stage rough–finish milling processes.
Experimental validation on polymer workpieces demonstrated the practical potential of the proposed
optimization framework under the tested conditions. Compared with CAM-recommended parameters,
the balanced Pareto-optimal solution reduced machining time from 5 min 58.52 s to 2 min 26.47 s,
corresponding to an approximate 59% reduction in machining time, while simultaneously decreasing
the peripheral cutting force from 698.18 N to 382.73 N, corresponding to an approximate 45%
reduction. In addition, the optimized strategy reduced the over-machining error observed in the CAM-
based approach and achieved a pocket depth closer to the target geometry. These results indicate that
machining allowance optimization can contribute to improved efficiency and dimensional control in
rectangular pocket milling.
The minimum-energy Pareto solution achieved the lowest energy consumption, decreasing from 0.44
kWh for the CAM baseline to 0.267 kWh. However, this benefit was accompanied by a substantial
increase in cutting force to 1203.45 N, highlighting the mechanical risks associated with single-
objective energy minimization. This finding underscores the necessity of multi-objective optimization,
as extreme solutions may compromise tool life, dimensional stability, and machine reliability despite
energy savings.
Overall, the proposed framework provides a rational and quantitative method for selecting machining
allowance and cutting parameters based on explicit performance targets. Rather than relying only on
default CAM recommendations, process planners may use Pareto-optimal solutions to balance
productivity, energy efficiency, cutting force, and dimensional accuracy according to specific
machining requirements. However, the present validation was limited to polymer workpieces and a
limited number of experimental trials. Future work should extend the method to metallic materials,
additional pocket geometries, repeated experiments, and quantitative surface roughness measurements
to further evaluate the robustness and general applicability of the proposed approach.
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5. Authorship acknowledgments
Tran Thanh Tung: Conceptualization; Methodology; Investigation; Data Analysis; Writing – Original
Draft. Nguyen Thi Anh: Formal Analysis; Investigation; Data Analysis. Nguyen Xuan Quynh:
Resources; Data Analysis. Tran Vu Minh: Conceptualization; Validation; Writing – Review and
Editing.
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