Cuckoo search algorithm for the selection of optimal ... - Springer Link

Int J Adv Manuf Technol (2013) 64:55–61 DOI 10.1007/s00170-012-4013-7

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 introduced 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, hybrid particle swarm algorithm, genetic algorithm, feasible direction 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 operational cost, and the quality of machining are often considered. The main goal in machining operations is to produce products with low costs and high quality. In order to manufacture the highest quality products, current optimization techniques must be improved. For many decades, the selection of optimal manufacturing parameters is a major issue faced every day in industry. Different optimization techniques have been used for A. R. Yildiz (*) Department of Mechanical Engineering, Bursa Technical University, Bursa, Turkey e-mail: [email protected]

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 introduce new optimization approaches to manufacture the products economically. Since population-based optimization techniques such as genetic algorithm, differential evolution, particle swarm optimization algorithm, and immune algorithm are more effective 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 evolutionarybased 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 continuum 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 twostep process is presented. Given an extended design domain with boundary and loading conditions, the method simultaneously optimizes the topology and geometry of an entire structure and the location and configuration of joints considering 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 evolutionary 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

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Int J Adv Manuf Technol (2013) 64:55–61

annealing (PGSA), to improve GSA’s computation performance and to find optimal machining parameters for multipass milling operations. The results showed that PGSA was more effective to optimize the cutting parameters for multipass 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], Taguchi 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 superiority 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 optimization problems and then applied them to the multiobjective 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 optimization 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 parameters 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 algorithm[29] and handbook recommendations [32].

Ci (i01–8) cl , c 0 cm, cmat, ct Cu d e F f Fc, Fc(per) FF, FR, FT

G, g K Ki (i01–3) Kp la m N n P, Pm Pr Q R

Ra, Ra(at) Sp T, Tu tm, ts, ttc V, Vhb, Vopt

2 Nomenclature The notation used in the machining model is defined as follows: A a, arad C ca

Chip cross-sectional area (square millimeter) Axial depth of cut, radial depth of cut (millimeter) Constant in cutting speed equation Clearance angle of the tool (degrees)

w W z

Coefficients carrying constants values Labor cost, overhead cost (dollar per minute) Machining cost, cost of raw material per part, cost of a cutting tool (dollar) Unit cost (dollar) Cutter diameter (millimeter) Machine tool efficiency factor Feed rate (millimeter per minute) Feed rate (millimeter per tooth) Cutting force, permitted cutting force (Newton) Feed, radial, and tangential forces resulting from all active cutting teeth (Newton) Slenderness ratio, exponent of slenderness ratio Distance to be traveled by the tool to perform the operation (millimeter) Coefficients carrying constant values Power constant depending on the workpiece material Lead (corner) angle of the tool Number of machining operations required to produce the product Spindle speed (revolution per minute) Tool life exponent Required power for the operation, motor power (kilowatt) Total profit rate (dollar per min) Contact proportion of cutting edge with workpiece per revolution Sale price of the product excluding material, setup, and tool changing costs (dollar) Arithmetic value of surface finish, and attainable surface finish (micrometer) Sale price of the product (dollar) Tool life (minute), unit time (minute) Machining time, setup time, tool changing time (minute) Cutting speed, recommended by handbook, optimum (meter per minute) Exponent of chip cross-sectional area Tool wear factor 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


57

is usually predetermined by the work piece geometry and operation sequence. It is recommended to machine the features 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 determining the proper cutting speed and feed rate combination [10]. The mathematical model of Rad and Bidhendi [10] is used in this paper.

Objective function ; Generate initial population of host nests ; while (stop criterion) Get a Cuckoo randomly by Lévy flights; Evaluate its quality/fitness ; Choose a nest among (say j) randomly; if end Abandon a fraction ( ) of worse nests [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;

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 c0 Þts þ þ

m P

Pm i¼1

Fig. 1 Pseudocode of cuckoo search

Where C5 ¼

0:78Kp Wzarad a 60pdePm

ð5Þ 5

1

ðcl þ co ÞK1i V i fi 1

cti K3i Vi ð1=nÞ1 fi ½ðwþgÞ=n1 þ

i¼1

m P

ðcl þ c0 Þ

10

ð1Þ

30

i¼1

A-A 120

The unit time for producing of a part in multitool milling is defined as follows: Tu ¼ ts þ

m X

K1i Vi 1 fi 1 þ

i¼1

m X

tci

100

ð2Þ

80

i¼1

The total profit rate is defined as follow: Pr ¼

Sp Cu Tu

40

ð3Þ

60

30

80 12

20

R5

3.2 Constraints In order to maximize the profit rate, allowable range of cutting speed and feed rate are imposed restriction by constraints. The constraints taken into consideration in this paper are defined as follows [10].

Slot 1

1. Maximum machine power 2. Surface finish requirement 3. Maximum cutting force permitted by the rigidity of the tool

Pocket Slot 2 Step

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: C5 Vf 0:8 1:

ð4Þ

Fig. 2 An example part

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Table 1 Speed and feed rate limits Operation no.

Operation type

Speed limits (m/min)

Feed rate limits (mm/tooth)

1 2 3 4 5

Face milling Corner milling Pocket milling Slot milling1 Slot milling2

60–120 40–70 40–70 30–50 30–50

0.05–0.4 0.05–0.5 0.05–0.5 0.05–0.5 0.05–0.5

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 C8 0l/Fc(per), the cutting force constraints can be defined as C8 Fc 1;

ð12Þ

4 Lévy Flıghts as random walks 3.2.2 Surface finish The surface finish value for plain milling and end milling operations can be defined as: f2 4d and for face milling f Ra ¼ 318 tanðlaÞ þ cotðcaÞ

ð6Þ

Ra ¼ 318

Sn ¼

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: C6 f 2 1

ð8Þ

where, C6 ¼

318ð4dÞ1 RaðatÞ

ð9Þ

and for face milling C7 f 1;

ð10Þ

where C7 ¼

318½tanðlaÞ þ cotðcaÞ1 RaðatÞ

ð11Þ

Xn i¼1

Xi ¼ X1 þ X2 þ ::: þ Xn ¼

Xn1 i¼1

Xi þ Xn

ð13Þ

¼ Sn1 þ Xn

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 straightforward ways is to use the so-called Mantegna algorithm. In Mantegna’s algorithm, the step length S can be calculated by S¼

u

ð14Þ

jvj1=b

where, β is a parameter between [1, 2] interval and considered to be 1.5; u and v are drawn from normal distribution as

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) Table 2 Required machining operation

The randomization has important role in population-based algorithms. The Lévy flights as random walks can be described as follows [33, 39]. A random walk includes a series of consecutive random steps. A random walk can be defined as

u N 0; σ2u ; v N 0; σ2u

ð15Þ

Operation no

Operation type

Tool no

a (mm)

K (mm)

Ra (μm)

Fc (per)

1 2 3 4 5

Face milling Corner milling Pocket milling Slot milling Slot milling

1 2 2 3 3

10 5 10 10 5

450 90 450 32 84

2 6 5 1

156,449.4 17,117.74 17,117.74 14,264.78 14,264.78


59

Table 3 Tools data Tool no

Tool type

Quality

D (mm)

z

Price ($)

la

ca

1 2 3

Face mill End mill End mill

Carbide HSS HSS

50 10 12

6 4 4

49.50 7.55 7.55

45 0 0

5 5 5

where σu ¼

n

rð1þb Þ sinðpb=2Þ r½ð1þb Þ=2b2ðb1Þ=2

o1=b

; σv ¼ 1

ð16Þ

Studies show that the Lévy fights can maximize the efficiency 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 reproduction 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 simplicity, 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 Table 4 Comparison of the results for milling operation

Method Handbook [32] Method of feasible direction [10] Genetic algorithm [29] Ant colony algortihm [16] Hybrid particle swarm (PSRE) [27] Immune algorithm Hybrid ımmune algorithm [29] Cuckoo search (CS)

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 Natureinspired 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 ðtþ1Þ

generate new solutions xi for the ith cuckoo, a Lévy flight is performed using the following equation: ðtþ1Þ

xi

ðtÞ

¼ xi a S

ð17Þ

where a>0 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

Cu—Unit cost

Tu—Unit time (min)

Pr—Profit rate (min)

$18.36 $11.35 $11.11 $10.20 $10.90 $11.08 $10.91 $10.90

9.40 5.48 5.22 5.43 5.052 5.07 5.04 5.03

0.71 2.49 2.65 2.72 2.79 2.75 2.79 2.80

60

a solution’s component to be discovered. Therefore, a probability matrix is produced as 1 if rand < pa Pij ¼ ð18Þ 0 if rand pa 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) milling 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: Sp 0$25 cmat 0$0.50 co 0$1.45 per min cl 0$0.45 per min ts 02 min tct 00.5 min C033.98 for HSS tools w00.28 C0100.05 for carbide tool K p 02.24 W01.1 n00.15 for HSS tools n00.3 for carbide tool g00.14 Machine tool data: Type: vertical CNC milling machine Pm 08.5 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 features 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, particle 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 compared 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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