This document describes a study applying a cuckoo search algorithm (CS) to optimize machining parameters in milling operations. The CS algorithm is introduced as a new optimization technique for manufacturing problems. To demonstrate the effectiveness of CS, a milling optimization problem is solved and the results are compared to other algorithms. The results show that CS is an effective and robust approach for optimizing machining parameters in milling operations.

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Cuckoo search algorithm for the selection of optimal machining parameters
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2. ORIGINAL ARTICLE
Cuckoo search algorithm for the selection
of optimal machining parameters in milling operations
Ali R. Yildiz
Received: 26 December 2011 /Accepted: 15 February 2012 /Published online: 25 March 2012
# Springer-Verlag London Limited 2012
Abstract In this research, a new optimization algorithm,
called the cuckoo search algorithm (CS) algorithm, is intro-
duced for solving manufacturing optimization problems. This
research is the first application of the CS to the optimization of
machining parameters in the literature. In order to demonstrate
the effectiveness of the CS, a milling optimization problem was
solved and the results were compared with those obtained using
other well-known optimization techniques like, ant colony
algorithm, immune algorithm, hybrid immune algorithm, hy-
brid particle swarm algorithm, genetic algorithm, feasible di-
rection method, and handbook recommendation. The results
demonstrate that the CS is a very effective and robust approach
for the optimization of machining optimization problems.
Keywords Milling operation . Cuckoo search . Optimization
1 Introduction
In machining applications, three conflicting objectives
which are the maximum production rate, minimum opera-
tional cost, and the quality of machining are often consid-
ered. The main goal in machining operations is to produce
products with low costs and high quality. In order to man-
ufacture the highest quality products, current optimization
techniques must be improved.
For many decades, the selection of optimal manufactur-
ing parameters is a major issue faced every day in industry.
Different optimization techniques have been used for
optimization of machining parameters in literature [1–13].
Recent advancements in optimization area introduced new
opportunities to achieve better solutions for manufacturing
optimization problems. Therefore, there is a need to intro-
duce new optimization approaches to manufacture the prod-
ucts economically.
Since population-based optimization techniques such as
genetic algorithm, differential evolution, particle swarm op-
timization algorithm, and immune algorithm are more effec-
tive than the gradient techniques in finding the global
minimum, they have been preferred in many applications
of science [14–32]. The first and well-known evolutionary-
based technique introduced in literature is the genetic
algorithms. The genetic algorithm (GA) was developed by
Holland [18] and has been commonly used in engineering
applications [19–21].
For instance, Yildiz and Saitou [14] developed a novel
approach for multicomponent topology optimization of con-
tinuum structures using a multi-objective genetic algorithm to
obtain Pareto optimal solutions that exhibits trade-offs among
stiffness, weight, manufacturability, and assemble ability. In
[14], a method for synthesizing structural assemblies directly
from the design specifications without going through the two-
step process is presented. Given an extended design domain
with boundary and loading conditions, the method simulta-
neously optimizes the topology and geometry of an entire
structure and the location and configuration of joints consid-
ering structural performance, manufacturability, and assemble
ability. The developed approach is applied to multicomponent
topology optimization of a vehicle floor frame.
In order to optimize machining parameters, the evolu-
tionary methods have been modified or hybridized with
other optimization techniques. Wang et al. [2] modified their
genetic simulated annealing [31] approaches and presented
a new hybrid approach, named parallel genetic simulated
A. R. Yildiz (*)
Department of Mechanical Engineering,
Bursa Technical University,
Bursa, Turkey
e-mail: aliriza.yildiz@btu.edu.tr
Int J Adv Manuf Technol (2013) 64:55–61
DOI 10.1007/s00170-012-4013-7

3. annealing (PGSA), to improve GSA’s computation perfor-
mance and to find optimal machining parameters for multi-
pass milling operations. The results showed that PGSA was
more effective to optimize the cutting parameters for multi-
pass milling operation than conventional geometric
programming and dynamic programming method.
In our previous work [23], GA was hybridized with
Taguchi’s robust design approach to optimize machining
parameters for multi-pass turning operations. In [23], Tagu-
chi method was used to refine the range of design variables.
After redefining the range of design variables using Taguchi
method, two multipass turning problem were optimized with
the new range of design variables using the GA. The supe-
riority of the proposed approach (HRGA) resulted from
refining of the ranges for design variables. The results found
by the HRGA were better than those of scatter search, GA
and combination of simulated annealing and Hooke–Jeeves
pattern search for turning operations.
Yildiz [30] has hybridized an artificial immune algorithm
with a hill climbing local search algorithm to solve optimi-
zation problems and then applied them to the multi-
objective I-beam and machine tool spindle design and also
manufacturing optimization problems.
Although some improvements regarding optimization of
cutting parameters in machining operations have been
achieved, due to the complexity of machining parameters
with conflicting objective and constraints, machining opti-
mization problems still present a matter of investigation.
Therefore, in recent years, there has been a growing interest
in applying the new approaches to further improving the
performance of machining parameters.
In this study, the cuckoo search algorithm (CS) is used to
optimize cutting parameters in milling operations. The CS is
applied to the case study to optimize the machining param-
eters in milling operations.
The results obtained by the CS for milling operations
indicate that the CS is more effective to optimize the cutting
parameters for milling operations than the feasible direction
method [10], ant colony algorithm [16], hybrid particle
swarm [27], hybrid immune algorithm [29], genetic algo-
rithm[29] and handbook recommendations [32].
2 Nomenclature
The notation used in the machining model is defined as
follows:
A Chip cross-sectional area (square millimeter)
a, arad Axial depth of cut, radial depth of cut
(millimeter)
C Constant in cutting speed equation
ca Clearance angle of the tool (degrees)
Ci (i01–8) Coefficients carrying constants values
cl, c0 Labor cost, overhead cost (dollar
per minute)
cm, cmat, ct Machining cost, cost of raw material
per part, cost of a cutting tool (dollar)
Cu Unit cost (dollar)
d Cutter diameter (millimeter)
e Machine tool efficiency factor
F Feed rate (millimeter per minute)
f Feed rate (millimeter per tooth)
Fc, Fc(per) Cutting force, permitted cutting force
(Newton)
FF, FR, FT Feed, radial, and tangential forces
resulting from all active cutting teeth
(Newton)
G, g Slenderness ratio, exponent of
slenderness ratio
K Distance to be traveled by the tool
to perform the operation (millimeter)
Ki (i01–3) Coefficients carrying constant values
Kp Power constant depending on the
workpiece material
la Lead (corner) angle of the tool
m Number of machining operations
required to produce the product
N Spindle speed (revolution per minute)
n Tool life exponent
P, Pm Required power for the operation,
motor power (kilowatt)
Pr Total profit rate (dollar per min)
Q Contact proportion of cutting edge
with workpiece per revolution
R Sale price of the product excluding
material, setup, and tool changing
costs (dollar)
Ra, Ra(at) Arithmetic value of surface finish,
and attainable surface finish (micrometer)
Sp Sale price of the product (dollar)
T, Tu Tool life (minute), unit time (minute)
tm, ts, ttc Machining time, setup time, tool
changing time (minute)
V, Vhb, Vopt Cutting speed, recommended by
handbook, optimum (meter per minute)
w Exponent of chip cross-sectional area
W Tool wear factor
z Number of cutting teeth of the tool
3 Optimization model of multi-tool milling operations
Depth of cut, feed rate, and cutting speed have the greatest
effect on the success of a machining operation. Depth of cut
56 Int J Adv Manuf Technol (2013) 64:55–61

4. is usually predetermined by the work piece geometry and
operation sequence. It is recommended to machine the fea-
tures with the required depth in one pass to keep machining
time and cost low, when possible. Therefore, the problem of
determining machining parameters is reduced to determin-
ing the proper cutting speed and feed rate combination [10].
The mathematical model of Rad and Bidhendi [10] is used
in this paper.
3.1 Objective function
In the optimization of machining parameters for milling
operations, the purpose is to maximize the total profit rate.
The maximization of total profit rate is carried out according
to the two objective functions, which are unit production
time and unit production cost.
The unit cost is the sum of material cost, setup cost,
machining cost, and tool changing cost. The unit cost is
defined as follows [10]:
Cu ¼ cmat þ ðcl þ c0Þts þ
Pm
i¼1 ðcl þ coÞK1iVi
1
fi
1
þ
P
m
i¼1
ctiK3iVi
ð1=nÞ1
fi
ðwþgÞ=n
½ 1
þ
P
m
i¼1
ðcl þ c0Þ
ð1Þ
The unit time for producing of a part in multitool milling
is defined as follows:
Tu ¼ ts þ
X
m
i¼1
K1iVi
1
fi
1
þ
X
m
i¼1
tci ð2Þ
The total profit rate is defined as follow:
Pr ¼
Sp Cu
Tu
ð3Þ
3.2 Constraints
In order to maximize the profit rate, allowable range of
cutting speed and feed rate are imposed restriction by con-
straints. The constraints taken into consideration in this
paper are defined as follows [10].
1. Maximum machine power
2. Surface finish requirement
3. Maximum cutting force permitted by the rigidity of the tool
3.2.1 Power
The required machining power for the machining operation
must not exceed the maximum obtainable value of motor
power. Therefore, the power constraint can be defined as:
C5Vf 0:8
1: ð4Þ
Where
C5 ¼
0:78KpWzarada
60pdePm
ð5Þ
Objective function ;
Generate initial population of host nests ;
while (stop criterion)
;
(say j) randomly;
if
) of worse nests
Get a Cuckoo randomly by Lévy flights;
Evaluate its quality/fitness
Choose a nest among
end
Abandon a fraction (
[and build new ones at new locations via Lévy flights]
Keep the best solutions (or nests with quality solutions);
Rank the solutions and find the current best;
end while
Post process results and visualization;
Fig. 1 Pseudocode of cuckoo search
Slot 2
Pocket
Step
Slot 1
A-A
120
10
30
80
12
40
80
100
30 60
20 R5
5
Fig. 2 An example part
Int J Adv Manuf Technol (2013) 64:55–61 57

5. 3.2.2 Surface finish
The surface finish value for plain milling and end milling
operations can be defined as:
Ra ¼ 318
f 2
4d
ð6Þ
and for face milling
Ra ¼ 318
f
tanðlaÞ þ cotðcaÞ
ð7Þ
The required surface finish Ra, must not surpass the
maximum accessible surface finish Ra(at) under the existing
conditions. Therefore, the surface finish constraint for end
milling can be defined as:
C6f 2
1 ð8Þ
where,
C6 ¼
318ð4dÞ1
RaðatÞ
ð9Þ
and for face milling
C7f 1; ð10Þ
where
C7 ¼
318 tanðlaÞ þ cotðcaÞ
½ 1
RaðatÞ
ð11Þ
3.2.3 Cutting force
The total cutting force Fc that results from the machining
operation must not exceed the allowed cutting force Fc (per)
that the tool can resist. The permitted cutting force for each
tool has been taken into account as its maximum limit for
cutting forces. Therefore, considering C80l/Fc(per), the cut-
ting force constraints can be defined as
C8Fc 1; ð12Þ
4 Lévy Flıghts as random walks
The randomization has important role in population-based
algorithms. The Lévy flights as random walks can be de-
scribed as follows [33, 39]. A random walk includes a series
of consecutive random steps. A random walk can be defined
as
Sn ¼
Xn
i¼1
Xi ¼ X1 þ X2 þ ::: þ Xn ¼
Xn1
i¼1
Xi þ Xn
¼ Sn1 þ Xn
ð13Þ
where, Sn presents the random walk with n random steps and
Xi is the ith random step with predefined length. The last
statement means that the next state will only depend on the
current existing state and the motion or transition Xn. In fact,
the step size or length can vary according to a known
distribution. A very special case is when the step length
obeys the Lévy distribution; such a random walk is called
a Lévy flight or a Lévy walk.
From the implementation point of view, the generation of
random numbers with Lévy flights consists of two steps: the
choice of a random direction and the generation of steps,
which obey the chosen Lévy distribution. Although the
generation of steps is quite tricky, there are a few ways of
achieving this. One of the most efficient and yet straightfor-
ward ways is to use the so-called Mantegna algorithm. In
Mantegna’s algorithm, the step length S can be calculated by
S ¼
u
v
j j1=b
ð14Þ
where, β is a parameter between [1, 2] interval and consid-
ered to be 1.5; u and v are drawn from normal distribution as
u N 0; σ2
u
; v N 0; σ2
u
ð15Þ
Table 1 Speed and feed rate limits
Operation no. Operation type Speed limits
(m/min)
Feed rate limits
(mm/tooth)
1 Face milling 60–120 0.05–0.4
2 Corner milling 40–70 0.05–0.5
3 Pocket milling 40–70 0.05–0.5
4 Slot milling1 30–50 0.05–0.5
5 Slot milling2 30–50 0.05–0.5
Table 2 Required machining
operation Operation no Operation type Tool no a (mm) K (mm) Ra (μm) Fc (per)
1 Face milling 1 10 450 2 156,449.4
2 Corner milling 2 5 90 6 17,117.74
3 Pocket milling 2 10 450 5 17,117.74
4 Slot milling 3 10 32 - 14,264.78
5 Slot milling 3 5 84 1 14,264.78
58 Int J Adv Manuf Technol (2013) 64:55–61

6. where
σu ¼ r 1þb
ð Þ sin pb=2
ð Þ
r 1þb
ð Þ=2
½ b2 b1
ð Þ=2
n o1=b
; σv ¼ 1 ð16Þ
Studies show that the Lévy fights can maximize the effi-
ciency of the resource searches in uncertain environments. In
fact, Lévy flights have been observed among foraging patterns
of albatrosses, fruit flies and spider monkeys.
5 Cuckoo search algorithm
The CS is inspired by some species of a bird family called
cuckoo because of their special lifestyle and aggressive repro-
duction strategy. These species lay their eggs in the nests of
other host birds (almost other species) with amazing abilities
such as selecting the recently spawned nests and removing
existing eggs that increase hatching probability of their eggs.
On the other hand, some of host birds are able to combat this
parasite behavior of cuckoos and throw out the discovered
alien eggs or build their new nests in new locations. This
algorithm contains a population of nests or eggs. For simplic-
ity, the following representations are used, where each egg in a
nest represents a solution and a cuckoo egg represents a new
one. If the cuckoo egg is very similar to the host’s, then this
cuckoo egg is less likely to be discovered; thus, the fitness
should be related to the difference in solutions. The aim is to
employ the new and potentially better solutions (cuckoos) to
replace a not-so-good solution in the nests [34, 38].
For simplicity in describing the CS, the following three
idealized rules are utilized [34]: (a) each cuckoo lays one egg
at a time and dumps it in a randomly chosen nest; (b) the best
nests with high quality of eggs are carried over to the next
generations; and (c) the number of available host nests is
constant, and the egg, which is laid by a cuckoo, is discovered
by the host bird with a probability of pa in the range of [0, 1].
The later assumption can be approximated by the fraction pa
of the n nests are replaced by new ones (with new random
solutions). With these three rules, the basic steps of the CS can
be summarized as the pseudocode shown in Fig. 1.
This pseudocode provided in the book entitled Nature-
inspired meta-heuristic algorithms by [33] is a sequential
version, and each iteration of the algorithm consists of two
main steps, but another version of the CS, which is supposed
to be different and more efficient, is provided by [35]. This
new version has some differences with the book version as
follows:
In the first step according to the pseudocode, one of the
randomly selected nests (except the best one) is replaced by
a new solution, which is produced by random walk with
Lévy flight around the so far best nest, considering the
quality. But in the new version, all of the nests except the
best one are replaced in one step by new solutions. To
generate new solutions x
tþ1
ð Þ
i for the ith cuckoo, a Lévy
flight is performed using the following equation:
x
tþ1
ð Þ
i ¼ x
ðtÞ
i a S ð17Þ
where a0 is the step size parameter and should be chosen
considering the scale of the problem, is set to unity in the CS
[34] and decreases function as the number of generations
increases in the modified CS [35–39] . It should be noted
that in this new version, the solutions’ current positions are
used instead of the best solution so far as the origin of the
Lévy flight. The step size is considered as 0.1 in this work
because it results in efficient performance of algorithm in
our example. The parameter S is the length of random walk
with Lévy flights according to Mantegna’s algorithm as
described in Eq. (14).
In the second step, the pa fraction of the worst nests is
discovered and replaced by new ones. However, in the new
version, the parameter pa is considered as the probability of
Table 4 Comparison of the
results for milling operation Method Cu—Unit cost Tu—Unit time (min) Pr—Profit rate (min)
Handbook [32] $18.36 9.40 0.71
Method of feasible direction [10] $11.35 5.48 2.49
Genetic algorithm [29] $11.11 5.22 2.65
Ant colony algortihm [16] $10.20 5.43 2.72
Hybrid particle swarm (PSRE) [27] $10.90 5.052 2.79
Immune algorithm $11.08 5.07 2.75
Hybrid ımmune algorithm [29] $10.91 5.04 2.79
Cuckoo search (CS) $10.90 5.03 2.80
Table 3 Tools data
Tool no Tool type Quality D (mm) z Price ($) la ca
1 Face mill Carbide 50 6 49.50 45 5
2 End mill HSS 10 4 7.55 0 5
3 End mill HSS 12 4 7.55 0 5
Int J Adv Manuf Technol (2013) 64:55–61 59

7. a solution’s component to be discovered. Therefore, a prob-
ability matrix is produced as
Pij ¼
1 if rand pa
0 if rand pa
ð18Þ
where, rand is a random number in [0, 1] interval and Pij is
discovering probability for the jth variable of the ith nest.
Then, all of the nests are replaced by new ones produced by
random walks (point-wise multiplication of random step
sizes with probability matrix) from their current positions
according to quality.
In this paper, the CS algorithm is used to define the
optimal machining parameters for milling operations. As a
supplement to help readers to implement the CS correctly, a
demo version is provided in the paper by [35].
6 Case study for milling operatıon
In this case study, it is aimed that a part shown in Fig. 2 is to
be produced using computer numerical control (CNC) mill-
ing machine. At the same time, it is desired that optimum
machining parameters are found with the maximum profit
rate. Specifications of the machine, material, and constant
values are given below [10].
Constants:
Sp0$25
cmat0$0.50
co0$1.45 per min
cl0$0.45 per min
ts02 min
tct00.5 min
C033.98 for HSS tools
w00.28
C0100.05 for carbide tool
K p02.24
W01.1
n00.15 for HSS tools
n00.3 for carbide tool
g00.14
Machine tool data:
Type: vertical CNC milling machine
Pm08.5 kW, e095%
Material data:
Quality: 10 L50 leaded steel.
Hardness0225 BHN
The speed and feed rate limits used for the case study are
given in Table 1.
The part shown in Fig. 2 includes four machining fea-
tures which are step, pocket and two slots. To manufacture
the part, it is required five milling operations, listed in
Table 2, which are face milling, corner milling, pocket
milling, slot milling 1, and slot milling 2, respectively.
The tools used for each operation and the data for tools
are listed in Table 3. The aim is to find the optimum cutting
conditions of each feature in order to machine the part with
maximum profit rate. The number of objective function
evaluation used by the CS for optimization search process
is 3,000. From the comparison of best results given in
Table 4, it is seen that the maxization of the total profit rate
in milling operation is achived by the CS.
The comparison of the results obtained by the CS, against
other techniques such as immune algorithm, ant colony, par-
ticle swarm, GA, the feasible direction method and handbook
recommendations, is given in Table 4. Function evaluation
numbers are 20,000 and 15,000 to find optimal solutions for
GA, and immune algorithm, respectively. The CS also
improves the convergence rate by computing the best value
and maintaining the less function evaluations 3,000. It can be
seen that better results for the best computed solutions are
achieved for the milling optimization problem compared to
the feasible direction method [10], ant colony algorithm [16],
hybrid particle swarm [27], hybrid immune algorithm [29],
genetic algorithm [29] and handbook recommendations [32].
7 Conclusions
In this paper, the cuckoo search algorithm is presented and
successfully implemented to the optimization of machining
parameters in milling operations. Significant improvement
is obtained with the CS compared to the feasible direction
method, ant colony algorithm, immune algorithm, hybrid
particle swarm, hybrid immune algorithm, genetic algorithm
and handbook recommendations.
As can be seen from Table 4, the CS is performed
effectively on the optimization of machining parameters of
the milling operation problem finding better solutions com-
pared to other approaches in the literature. These results
show that the CS is an important alternative for optimization
of machining parameters in milling operations. In addition,
the CS is a generalized solution method so that it can be
easily employed to consider the optimization models of
milling regarding various objectives and constraints.
Other possible future works include application of the CS
to the other metal cutting problems such as turning, drilling,
grinding etc. operations in manufacturing industry as well as
design optimization problems.
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