The multi-pass turning operation is one of the most commonly used machining methods in manufacturing field. The main objective of this operation is to minimize the unit production cost. This paper proposes a Gaussian quantum-behaved bat algorithm (GQBA) to solve the problem of multi-pass turning operation. The proposed algorithm mainly includes the following two improvements. The first improvement is to incorporate the current optimal positions of quantum bats and the global best position into the stochastic attractor to facilitate population diversification. The second improvement is to use a Gaussian distribution instead of the uniform distribution to update the positions of the quantum-behaved bats, thus performing a more accurate search and avoiding premature convergence. The performance of the presented GQBA is demonstrated through numerical benchmark functions and a multi-pass turning operation problem. Thirteen classical benchmark functions are utilized in the comparison experiments, and the experimental results for accuracy and convergence speed demonstrate that, in most cases, the GQBA can provide a better search capability than other algorithms. Furthermore, GQBA is applied to an optimization problem for multi-pass turning, which is designed to minimize the production cost while considering many practical machining constraints in the machining process. The experimental results indicate that the GQBA outperforms other comparison algorithms in terms of cost reduction, which proves the effectiveness of the GQBA.
Bat algorithmquantum behaviorgaussian distributionnumerical optimizationmulti-pass turningIntroduction
In the manufacturing field, the technological challenge of machining operations is to produce products of the desired quality with high productivity and low cost. To reduce the machining cost for economical reasons, the optimization of the machining parameters is one of the most important issues, since these parameters have a strong impact on productivity, cost and quality [1]. Most turning processes require multi-pass turning, considering economic factors. The optimization problem for machining parameters in multi-pass turning becomes very complicated when a large number of actual machining constraints need to be considered [2]. Traditional optimization techniques, namely, dynamic programming [3] and the sequential unconstrained minimization technique [4], may be helpful in addressing some specific issues. However, these methods tend to find local optimal results. As a result, heuristic algorithms and nature-inspired swarm intelligence (SI) techniques have been introduced to solve economic machining problems because they have power in global search and robustness [5–9]. Chen proposed a scatter search to address optimization problems [10]. The genetic algorithm (GA) has been used extensively by some researchers [11–13] to optimize the process of multi-pass turning. The researchers applied varying improvements of the genetic operators to improve the GA performance to obtain better results. To solve optimization problems, ant colony optimization (ACO) [13–15], particle swarm optimization (PSO) [16–18], cuckoo optimization algorithm (COA) [9], grasshopper optimization algorithm (GOA) [19], bird swarm algorithm (BSA) [20], and the hybrid immune algorithm [21] were also developed.
In this paper, the first application of an improved version of the prevailing bat algorithm (BA) in the optimization of turning operations is presented. The BA is a recently proposed swarm intelligence optimization technique [22]. By mimicking the foraging behaviours of bats in searching for prey, the BA has incorporated the advantages of many classical techniques in a reasonable manner, including PSO, the GA [23] and simulated annealing (SA) [24]. Not only does the BA retain simplicity, it has also been shown to be more effective than its predecessors, especially in cases of lower dimensions. In addition, it is convenient to implement using a variety of programming languages. Therefore, the BA has been used to solve various engineering optimization applications [25]. However, since the population diversity of the BA is not sufficiently high, the BA may fall into a local optimum when dealing with applications of high dimensions [26]. To address this defect, many versions of the BA have been proposed, such as the CLBA [27], IBA [28], DLBA [29], and HSBA [30]. The quantum bat algorithm (QBA) [31] is a new technique based on the idea of quantum computing, which can increase the population diversity of the BA. In addition, considering the self-adaptive compensation of the Doppler effect in sound, the echolocation mechanism of bats can be simulated in the QBA. The QBA has been proven to outperform the standard BA and some well-known algorithms [31].
However, because the position of the stochastic attractor determines the central position of the quantum-behaved bat search area in the population and some critical random numbers are produced in the quantum mutation phase utilizing a uniform distribution, there is still premature convergence in the QBA to a certain extent. Motivated by these disadvantages, this paper proposes a QBA approach based on a Gaussian distribution (GQBA) to address the problem of premature convergence. The main contributions of this paper are as follows:
We propose an improved quantum-behaved bat algorithm using Gaussian distribution and apply it to the multi-pass turning problem.
We introduce a new stochastic attractor updating strategy to promote the diversification of swarms and a new quantum-behaved bat updating strategy to perform a more accurate search and avoid premature convergence.
Experiments on the numerical and multi-pass turning optimization indicate the efficiency of the GQBA. Comparisons with other comparative algorithms validate that the GQBA is competitive and is a good alternative approach.
The remainder of this paper is structured as follows: Section 2 briefly introduces related works. Section 3 gives the details of the GQBA proposed in this article. Section 4 presents the numerical validation and comparison. Section 5 presents an application of the GQBA to the multi-pass turning operation problem as well as the experimental results. Finally, the concluding comments and some future research directions are presented in Section 6.
Related WorksThe Original BA Algorithm
When bats are foraging for food, they use echolocation to find prey and avoid obstacles. Inspired by the foraging behaviour of real bats, Yang proposed the bat algorithm in 2010 [22]. The original BA uses a frequency-tuning method to promote the diversification of the population. Additionally, it utilizes the automatic scaling technique to maintain a balance between exploration and exploitation during the optimization process by imitating the changes in pulse loudness and emission frequency during foraging. The procedures of the original BA are given below.
Each virtual bat in the swarm moves towards the global optimal position, which is the reason that all bats fly towards prey when foraging. The frequency (fi), velocity (vi) and position value (xi) of the virtual bats change during the search procedure according to the following Eqs. (1)–(3):
fi=fmin+(fmax−fmin)βvit=vit−1+(xit−1−gbt−1)fixit=xit−1+vitwhere β is a stochastic value that is produced uniformly in [0, 1], fmin denotes the lowest frequency, fmax denotes the highest frequency, and gbt indicates the global optimal position. Utilizing the formulas above, the BA can carry out the exploration operation.
For the exploitation stage, to generate a new candidate position for every virtual bat when a position is selected from the current optimal positions, a technique called a local random walk is adopted, which is given as Eq. (4):
xnew=xold+εA¯t−1,where ε indicates a uniform stochastic value in [−1, 1] that determines the direction of the new candidate position. Note that A¯t−1 indicates the mean of the loudness values of all virtual bats at the (t−1)th iteration.
When foraging, every virtual bat gradually adjusts its loudness value and emission frequency to locate prey. The equations for updating the value of loudness Ai and the value of emission frequency ri in every iteration can be given as Eqs. (5) and (6):
Ait=αAit−1rit=ri0[1−exp(−γ(t−1))],where ri0 denotes the initial value of the emission rate of the ith virtual bat, α and γ indicate the loudness attenuation coefficient and the rate enhancement coefficient of pulse emission, respectively. The value of α is in the range [0, 1], and the value of γ is positive (γ>0). Both α and γ are constants. In fact, similar to the cooling parameter in SA, the parameter α determines the convergence characteristic of the BA. α=γ is commonly used in the literature.
The pseudo-code of the original BA is shown in Fig. 1.
Pseudo-code of the original BA [<xref ref-type="bibr" rid="ref-22">22</xref>]The QBA Algorithm
The original BA has been used in many applications. However, since the population diversity of the BA is not sufficiently high, the BA does not perform well in multimodal cases. By analyzing the flight path of virtual bats, Meng et al. [31] developed the QBA. In the QBA, both the quantum behaviour and Doppler effect are taken into account, as well as other characteristics of the original BA. The mutation operator of quantum behaviour contributes to promoting the diversity of the swarm, which is ultimately beneficial in avoiding premature convergence.
The QBA is generally proposed on the framework of the standard BA. The exploration and exploitation procedures in the QBA are controlled by the decreasing loudness value A and increasing emission rate value r, respectively. However, the approach to generating new candidate positions in the QBA is not the same as that in the standard BA. Two more idealized rules have been introduced [32]: (1) the virtual bats have two foraging habitats instead of one foraging habitat depending on a random choice, and (2) the virtual bats self-adaptively compensate for the Doppler effect in sound. The positions of the virtual bats determined by quantum behaviour in the QBA can be described as Eq. (7):
xidt={pdt−1+δ×|mbestd−xidt−1|×ln(1u),z<0.5pdt−1−δ×|mbestd−xidt−1|×ln(1u),z≥0.5where
pdt−1=gbdt−1δ=δmax−(δmax−δmin)tmax×tmbestd=1N∑i=1Npbestid
pdt−1 denotes the value in the dth dimension of the best previous position gb at iteration t, called the stochastic attractor, xidt indicates the position of the ith virtual bat in the dth dimension at the tth iteration, and u and z are random values generated uniformly from [0, 1]. δ is a design parameter named the contraction-expansion parameter [32], which can be utilized to control the convergence rate of the techniques. δmax and δmin indicate the initial value and final value of δ, respectively. In the QBA, δmax=1 and δmin=0.5 are adopted. mbest, called the (Mean Best), represents the global point of the swarm, which is also the average of the present optimal positions pbest of all virtual bats. N is the population size, and pbesti,d indicates the value of the present optimal position of the ith bat in dimension d.
Considering the self-adaptive capability of virtual bats to compensate for the Doppler effect, the equations mentioned in Eqs. (1) and (2) can be changed as Eqs. (11)–(13):
fid=(c+vit)(c−vgt−1)×fid×[1+ϕi×(gbdt−xidt)|gbdt−xidt|+ε]vidt=(w×vidt−1)+(gbdt−xidt)×fidxidt=xidt−1+vidt,where fid is the frequency value of the ith virtual bat in the dth dimension, vgt−1 indicates the velocity value of the global optimal position at the (t−1)th iteration, and ϕi indicates a positive value of the ith virtual bat in [0, 1]. The inertia weight factor w is used to update the velocity vector, which is similar to the inertia weight factor in PSO. C (c=340 m/s) is the speed of sound in the air.
In the exploitation stage, in the QBA, the new candidate position of the virtual bat is produced as Eqs. (14) and (15):
xidt=gbdt−1×χσ2=|Ait−1−A¯t−1|+ε,where χ=N(0,σ2) indicates a Gaussian distribution, xidt denotes the position of the ith virtual bat at the tth iteration and gbdt indicates the current global optimal position searched by the virtual bats in the dth dimension. Ait−1 indicates the loudness value of the ith virtual bat at the (t−1)th iteration. ϵ is utilized here to guarantee that σ2 is positive.
During the search process, if the fitness of the objective function is not promoted in the Kth time step, then a simple strategy is employed to enhance the algorithm, which is to reassign the loudness value Ai and reset the temporary emission rate value ri, which is a stochastic value generated uniformly from [0.85, 0.9].
The pseudo-code of the QBA is given in Fig. 2.
Pseudo-code of the QBA [<xref ref-type="bibr" rid="ref-31">31</xref>]The Proposed GQBA
As mentioned above, the QBA is an improved BA variant that has good performance. However, by analyzing Eq. (7), it can be seen that due to the convergence of quantum-behaved bats to the stochastic attractor pdt−1, when the stochastic attractor is very close to the global optimal individual, the bats will be concentrated near the global optimal solution, resulting in a mass aggregation phenomenon. However, if the stochastic attractor is located at the local optimal solution and is far from the global optimal solution, the bats will be distributed near the local optimal solution with a higher probability, which can easily lead to the premature convergence of the algorithm. Thus, the position of the stochastic attractor pdt−1 determines the central position of the quantum-behaved bats’ search area in the population.
To address the premature convergence problem of the QBA, this paper makes full use of the guiding role of the stochastic attractor and proposes a new strategy to update the position of the stochastic attractor as well as the bats in the population; this leads to the quantum-behaved bat algorithm based on a Gaussian distribution (the GQBA). In the implementation of the GQBA, some improved schemes have been substituted into the QBA, which are illustrated in the remainder of this section.
The first modification is to update the stochastic attractor with the guidance of the current optimal position value of the ith bat pbestidt−1 and the global best position gbdt−1. In fact, the introduction of pbestidt−1 can increase the diversity of the population and lead the QBA to perform a more thorough global search. Parameter pdt−1 of Eq. (7) is changed according to the Eq. (16):
pdt−1=c1pbestidt−1+c2gbdt−1c1+c2,where c1 and c2 are generated randomly within [0, 1].
The second change is to substitute coefficients u into Eq. (7) with the absolute value of the Gaussian distribution whose σ2 is 1 and mean is 0, which can be defined as G=abs(N(0,1)). Many studies [33–35] have shown that a Gaussian distribution with a long tail can perform a more accurate search near the last generation of individuals, improve the local search capability, provide a greater search step and random walking distance, expand the search space, and improve the algorithm’s ability to jump out of local optimality.
The formula of probability density for G in one dimension can be stated as Eq. (17):
f(x)=22πexp(−x22),x≥0.
Therefore, in the GQBA, the position of the quantum-behaved bats can be described as Eq. (18):
xidt={pdt+δ×|mbestd−xidt|×ln(1G),rand<0.5pdt−δ×|mbestd−xidt|×ln(1G),rand≥0.5,where G=abs(N(0,1)). Clearly, in line with Eq. (17), f(0)=0; hence, G is within the logarithmic function’s domain range (>0).
In the GQBA, the stochastic attractor in terms of pd instructs the exploration stage to ensure the convergence rate, while the habitat selection design of two different foraging habitats (a Gaussian quantum-behaved habitat and a Doppler effect-compensated mechanical habitat) contributes to the exploitation stage to help the algorithm jump out of the local best positions and avoid premature convergence.
The procedure of the GQBA is shown in Fig. 3.
Pseudo-code of the GQBANumerical Validation and Comparison
In this section, thirteen classical benchmark functions are utilized to verify the efficiency of the GQBA, which are illustrated in Tables 1 and 2. These benchmark functions have been utilized in many numerical optimization studies [36–39]. In this paper, the thirteen classical test functions can be divided into two classes. The first class consists of 7 unimodal benchmark functions with only one global best solution and effectively validates the meta-heuristic algorithms in terms of the convergence rate as well as the local search ability. The second class consists of 6 multimodal functions, the quantity of whose local optima increases exponentially. Multimodal functions are usually utilized to examine the global search ability of algorithms. All the tests on each benchmark function are repeated 30 times independently. All the tests described in this article were executed using MATLAB 2014a on a computer equipped with an Intel(R) Core(TM) i5-6500 CPU at 3.20 GHz with 8.0 GB of RAM.
Unimodal benchmark test cases <inline-formula id="ieqn-60"><mml:math id="mml-ieqn-60"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>–<inline-formula id="ieqn-61"><mml:math id="mml-ieqn-61"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>7</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>
Instance
D
Range
fmin
F1(x)=∑i=1nxi2
30
[−100, 100]
0
F2(x)=∑i=1n|xi|+∏i=1n|xi|
30
[−10, 10]
0
F3(x)=∑i=1n(∑j=1ixj)2
30
[−100, 100]
0
F4(x)=maxi{|xi|,1≤i≤n}
30
[−100, 100]
0
F5(x)=∑i=1n−1[100(xi+1−xi2)2+(xi−1)]2
30
[−30, 30]
0
F6(x)=∑i=1n([xi+0.5])2
30
[−100, 100]
0
F7(x)=∑i=1nixi4+random[0,1]
30
[−1.28, 1.28]
0
Multimodal benchmark test cases <inline-formula id="ieqn-70"><mml:math id="mml-ieqn-70"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>8</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>–<inline-formula id="ieqn-71"><mml:math id="mml-ieqn-71"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>
Since the heuristic algorithm is a random optimization method, at least 10 independent runs need to be carried out to generate meaningful statistical results. In addition to the average value and standard deviation, statistical tests, such as the Wilcoxon rank sum test, are required to test the significance of the consequences based on each separate run. In this research, nonparametric Wilcoxon rank sum tests were carried out to examine whether the consequences of the GQBA were conspicuously different from those of comparative techniques. p<0.05 indicates that there exists a conspicuous difference between two techniques, while p≥0.05 indicates that there is no conspicuous difference.
To examine the efficiency of the GQBA and to obtain an exhaustive comparison, the GQBA and four other techniques, the BA [22], the QBA [31], PSO [40] and the GSA [41], are tested on the sets of benchmark instances above. In all experiments, the total number of iterations is 10000, and the number of individuals is 50. Table 3 shows the other specific parameter designs for each technique. The simulation results are illustrated in Tables 4 and 5 and Figs. 4–16. The best consequences for each test function are denoted in bold.
The parameter design for the BA, QBA, GQBA, PSO and GSA
Note: u(m, n) represents a random value generated uniformly from [m, n].
Numerical comparison of different methods on the test instances
F
BA
QBA
GQBA (Ours)
PSO
GSA
Mean
SD
Mean
SD
Mean
SD
Mean
SD
Mean
SD
F1
3.81e−05
5.72e−05
2.20e−36
1.12e−35
0
0
9.48e−84
5.19e−83
8.17e−18
1.68e−18
F2
1.96e+04
7.70e+04
2.36e−21
5.49e−21
0
0
1.41e−40
6.08e−40
1.33e−08
1.57e−09
F3
1.57e−04
6.44e−05
1.48e−19
5.41e−19
0
0
6.09e−08
7.02e−08
3.62e−17
9.34e−18
F4
3.99e+01
6.35e+00
1.52e+01
5.63e+00
0
0
1.03e−04
1.07e−04
1.59e−09
1.74e−10
F5
8.61e+00
3.00e+01
2.23e+01
1.97e+00
1.99e+01
1.12e+01
2.78e+01
2.58e+01
1.42e+01
3.04e−01
F6
2.73e−05
7.57e−06
1.43e−01
3.00e−01
2.95e−15
2.25e−15
5.34e−33
7.19e−33
7.51e−18
2.24e−18
F7
1.89e−03
5.94e−04
1.54e−03
9.24e−04
1.29e−05
1.15e−05
4.14e−03
1.28e−03
1.10e−02
2.72e−03
F8
−7.19e+03
8.78e+02
−6.40e+03
8.72e+02
−6.90e+03
8.76e+02
−6.87e+03
8.52e+02
−2.71e+03
5.20e+02
F9
1.89e+02
3.55e+01
1.47e+02
5.79e+01
1.04e+01
1.37e+01
2.25e+01
6.11e+00
1.25e+01
3.22e+00
F10
20.0e+02
1.15e−03
6.28e−01
1.07e+00
1.13e−15
9.01e−16
9.53e−15
3.19e−15
2.24e−09
2.59e−10
F11
1.74e+02
5.06e+01
7.51e−03
1.08e−02
0
0
1.54e−02
1.25e−02
9.04e−04
3.80e−03
F12
2.46e+01
1.21e+01
4.72e+00
5.26e+00
2.97e−01
9.07e−01
1.60e−32
4.51e−34
4.97e−20
1.35e−20
F13
7.28e+01
9.42e+00
9.75e+00
1.28e+01
2.76e−15
2.49e−15
3.66e−04
2.01e−03
7.89e−19
1.75e−19
<italic>p</italic>-values over all tests
F
GQBA
BA
QBA
PSO
GSA
F1
N/A
1.2118e − 12
1.2118e − 12
1.2118e − 12
1.2118e − 12
F2
N/A
1.2118e − 12
1.2118e − 12
1.2118e − 12
1.2118e − 12
F3
N/A
1.2118e − 12
1.2118e − 12
1.2118e − 12
1.2118e − 12
F4
N/A
1.2118e − 12
1.2118e − 12
1.2118e − 12
1.2118e − 12
F5
N/A
0.0042
0.0011
0.0011
3.9881e − 04
F6
N/A
3.0199e − 11
3.0199e − 11
3.0199e − 11
3.0199e − 11
F7
N/A
3.0199e − 11
3.0199e − 11
3.0199e − 11
3.0199e − 11
F8
N/A
0.2838
0.0339
0.7958
3.0199e − 11
F9
N/A
1.6179e − 11
4.4870e − 11
0.0045
0.0619
F10
N/A
2.3638e − 12
2.3547e − 12
5.9197e − 13
2.3638e − 12
F11
N/A
1.2118e − 12
2.9343e − 05
6.1501e − 10
0.1608
F12
N/A
4.0772e − 11
9.7555e − 10
2.0535e − 11
3.0199e − 11
F13
N/A
3.0199e − 11
3.0199e − 11
4.6152e − 10
3.0199e − 11
Note: N/A represents ‘Not Applicable’, and p values that are not less than 0.05 are underlined.
Convergence comparison of five techniques for <inline-formula id="ieqn-137"><mml:math id="mml-ieqn-137"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-138"><mml:math id="mml-ieqn-138"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-139"><mml:math id="mml-ieqn-139"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>3</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-140"><mml:math id="mml-ieqn-140"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>4</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-141"><mml:math id="mml-ieqn-141"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-142"><mml:math id="mml-ieqn-142"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>6</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-143"><mml:math id="mml-ieqn-143"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>7</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-144"><mml:math id="mml-ieqn-144"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>8</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-145"><mml:math id="mml-ieqn-145"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>9</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-146"><mml:math id="mml-ieqn-146"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>10</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-147"><mml:math id="mml-ieqn-147"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>11</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-148"><mml:math id="mml-ieqn-148"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>12</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>Convergence comparison of five techniques for <inline-formula id="ieqn-149"><mml:math id="mml-ieqn-149"><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mn>13</mml:mn></mml:mrow></mml:msub></mml:math></inline-formula>
As shown in Table 4, the GQBA outperforms the other techniques in most cases, followed by PSO, the GSA, the QBA and the BA. To a certain extent, this is evidence that when the dimensionality of the search scope is high, the local optimal avoidance effect of the BA is insufficient.
As seen from Table 4, concerning accuracy, the GQBA has a better mean than the other techniques in eight out of 13 instances (F1, F2, F3, F4, F7, F9, F10 and F11). Meanwhile, in terms of stability, the GQBA performs better than the other techniques in 7 out of 13 instances (F1, F2, F3, F4, F7, F10 and F11). For local optimum avoidance, with regard to the p values in Table 5, the performance of the GQBA in three out of 13 test functions is not significantly different (F8, F9 and F11). Taking into account the accuracy results above, it can be concluded that the GQBA developed in this paper is able to provide significant results in six out of 13 instances (F1, F2, F3, F4, F7 and F10), which means that the GQBA has better local optimum avoidance. For the other functions, the performance of the GQBA is ranked second and third two and three times, respectively. In general, this means that the GQBA performs well for both types of functions.
Figs. 4–16 show the convergence comparison of the five algorithms for the thirteen benchmark cases. The values shown in the convergence curves above them indicate the average of the objective values obtained from 30 separate runs. On the basis of the figures, the original BA and QBA converge at a faster speed with fewer iterations. However, in many cases, they are more likely to fall into local optima. It can also be seen from the figures that the convergence rate of the GQBA is similar to that of the QBA, while the GQBA can prevent premature convergence and can offer higher performance accuracy on most benchmark instances.
GQBA for Optimization of the Multi-Pass Turning Process
In this section, the GQBA with the pruning strategy is developed to address the constrained manufacturing optimization problem of multi-pass turning optimization. The goal of this problem is to find the optimal cutting parameters to minimize the unit production cost (UC) [8,9,42]. The minimization process is subject to many machining constraints that describe the states of the machining procedure. Machining optimization has been investigated by different techniques, such as ACO [14], PSO [16,43], the GA [5], the artificial bee colony (ABC) [44–46], the COA [9], the firefly algorithm (FA) [47], the flower pollination algorithm (FPA) [8], and differential evolution (DE) [48]. It is very convenient to compare the proposed GQBA with the methods developed previously.
Mathematical Model of Multi-Pass Turning
In this article, we adopt the mathematical model presented in [8,9,42] for optimizing the cutting parameters. The goal of the optimization model is to find the optimal cutting parameters, i.e., the depth-of-cut, feed rate and cutting speed, for both finishing and rough machining to minimize the unit production cost. A schematic representation of a turning operation is given in Fig. 17.
Schematic representation of a multi-pass turning operationThe Objective Function: Unit Production Cost (UC)
The UC of the multi-pass turning process is usually indicated by a combination of 4 basic cost factors:
The cost due to the real cutting time (CM).
The machine idle cost for setup operations and tool idling motion (CI).
The cost of tool replacement (CR).
The tool cost (CT).
Thus, the UC can be expressed as following Eq. (19) [8,9,42]:
UC=CM+CI+CR+CT=[πDL1000Vrfr(dt−dsdr)+πDL1000Vsfs]k0+[tc+(h1L+h2)(dt−dsdr+1)]k0+[πDL1000Vrfr(dt−dsdr)+πDL1000Vsfs]teTpk0+[πDL1000Vrfr(dt−dsdr)+πDL1000Vsfs]ktTp
Machining Condition Constraints
To minimize the UC, practical machining constraints that describe the states of the machining procedure, i.e., finishing and rough machining, are taken into account. They are described in detail as follows [8,42]:
Rough Machining
The constraints of rough machining are listed as Eqs. (20)–(27):
Range of cutting speed:VrL≤Vr≤VrURange of feed rate:frL≤fr≤frURange of depth-of-cut:drL≤dr≤drUTool life constraint:TL≤Tr≤TUMaximum cutting force:Fr=k1frμdrν≤FUPower constraint:Pr=k1frμdrνVr6120η≤PU
The chip-tool interface temperature constraint is expressed as Eq. (26):
Qr=k2Vrτfrϕdrδ≤QUStable cutting region constraint:Vrλfrdrυ≥SC
Finish Machining
The constraints of finish machining are listed as Eqs. (28)–(36):
Range of cutting speed:VsL≤Vs≤VsURange of feed rate:fsL≤fs≤fsURange of depth-of-cut:dsL≤ds≤dsUTool life constraint:TL≤Ts≤TUMaximum cutting force:Fs=k1fsμdsν≤FUPower constraint:Ps=k1fsμdsνVs6120η≤PU
The chip-tool interface temperature constraint is expressed as Eq. (26):
Qs=k2Vsτfsϕdsδ≤QUStable cutting region constraint:Vsλfsdsυ≥SCSurface finishing constraint:fs28R≤SRU
Parameter Relations
The practical relationship between finish and rough machining can be given by Eqs. (37)–(40):
Vs≥k3Vrfr≥k4fsdr≥k5dsn=dt−dsdr,andn∈Z+where the tool life can be formulated as Eq. (41).
T=C0Vpfqdr
It is supposed that the same tool is utilized during the whole machining operation procedure for both finishing and roughing. The wear rate of the cutter tools is commonly different between finishing and roughing due to changing machining conditions. Therefore, the life of the tool can be calculated as Eq. (42):
Tp=θTr+(1−θ)Ts,θ∈[0,1]where
Tr=C0VrpfrqdrrTs=C0Vspfsqdsr
In some previous research work, a simplified formula for Tp (Eq. (42)) was adopted by ignoring the weight factor θ, as given by Eq. (45):
Tp=Tr+Ts.
Pruning Strategies Using the Theoretical Lower Bound of the Subproblem
The machining parameters of multi-pass turning that need to be optimized include a single finish cut and multiple rough cuts. Therefore, the available quantity of rough cuts should be restricted to certain ranges by Eq. (46):
nL≤n≤nU,where nL=⌈(dt−dsU)/drU⌉, nU=⌊(dt−dsL)/drL⌋ and n is a fixed integer.
Therefore, the total quantity of available values of n can be given as Eq. (47):
m=(nU−nL+1),where m is generally a small integer.
Hence, the issue of optimizing the cutting parameters of multi-pass turns is divided into m subproblems. Therefore, the search process for the entire optimization problem is equivalent to search processes of the m subproblems, and the solution corresponding to the optimal fitness value among the m subproblems is taken as the solution of the entire optimization problem. Inspired by pruning strategies, we have found that the theoretical lower bound on the unit production cost for each subproblem can be used to reduce the total running time during the enumerative process, as reported in our previous work [49]. The theoretical lower bound on the unit production cost can be calculated according to the different quantities of rough cuts for each subproblem [49], and the theoretical lower bound on UC for the j-th subproblem is represented by UCjL. The flowchart of the GQBA for multi-pass turning optimization is given in Fig. 18.
The flowchart of the GQBA for multi-pass turning optimization
The various notation used above is defined in Fig. 19.
List of symbols [<xref ref-type="bibr" rid="ref-48">48</xref>]Experimental Verification and Comparisons
All the simulations were performed on a PC with the same characteristics as in Section 4. The numerical validation and comparisons using the machining model data [8,42] are shown in Table 6. We note that the tool life is usually defined by two different expressions (Eqs. (42) and (45)); thus, the proposed GQBA was tested using these two definitions. In each case, the GQBA was performed 50 times to obtain the average solution and optimized solution. The total number of iterations and the total number of individuals in each test were set to 7500 and 100, respectively. The remaining parameter design of the GQBA is given in Table 3.
Data for the machining model
Parameter
Value
Parameter
Value
Parameter
Value
Parameter
Value
D
50 mm
L
300 mm
dt
6 mm
VrL
50 m/min
frL
0.1 mm/rev
drL
1 mm
VrU
500 m/min
frU
0.9 mm/rev
drU
3 mm
VsL
50 m/min
fsL
0.1 mm/rev
dsL
1 mm
VsU
500 m/min
fsU
0.9 mm/rev
dsU
3 mm
p
5
q
1.75
r
0.75
μ
0.75
ν
0.95
η
0.85
λ
2
υ
−1
τ
0.4
ϕ
0.2
δ
0.105
R
1.2 mm
C0
6×1011
TL
25 min
TU
45 min
FU
200 Kgf
PU
5 kW
SC
140
QU
1000°C
SRU
10 μm
h1
7×10−4
h2
0.3
te
1.5 min/edge
tc
0.75 min/piece
kt
2.5 $/edge
k0
0.5 $/min
k1
108
k2
132
k3
1
k4
2.5
k5
1
Table 7 presents the results of the average unit production costs obtained by the GQBA using various mathematical models with different tool life expressions and cutting depths. We note that none of the solutions obtained in the experiment violate the practical machining constraints, suggesting that they are feasible. Moreover, it can be seen from this table that due to the low standard deviation, the solution obtained in each instance fluctuates within a small range, which indicates that the GQBA presented in this paper has good stability.
Results obtained by the GQBA
Model
Depth-of-cut (mm)
Average UC ($/piece)
Standard deviation
Function evaluations
Execution time (sec/run)
Tp=Tr+Ts
6
1.9602
0.00113
750, 000
95
Tp=Tr+Ts
8
2.4398
0.00153
750, 000
96
Tp=θTr+(1−θ)Ts
6
2.0284
0.00050
750, 000
98
Tp=θTr+(1−θ)Ts
8
2.5514
0.00172
750, 000
99
The optimization problem has been investigated by various approaches. To demonstrate the effectiveness of the proposed GQBA, we compare the simulation results with those reported in recent literature [5,8,9]. For a fair comparison, we select the results of other algorithms with consistent common parameters in these literatures. The detailed comparison results are summarized in Tables 8–11, where the best results are indicated by underlines.
Comparison of the experimental results among different algorithms (when <inline-formula id="ieqn-206"><mml:math id="mml-ieqn-206"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, <inline-formula id="ieqn-207"><mml:math id="mml-ieqn-207"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula> mm)
Algorithm
Vr(m/min)
Vs(m/min)
fr(mm/rev)
fs(mm/rev)
dr(mm)
ds(mm)
UC($/piece)
Constraint violation
GQBA (Ours)
123.3238
169.9843
0.5654
0.2261
3
3
1.9592
0
FPA [8]
123.3431
169.9785
0.5655
0.2262
3
3
1.9591
0
COA [9]
123.1462
169.9876
0.5655
0.2262
3
3
1.959
0
HPSO [43]
123.3424
169.9783
0.5655
0.2262
3
3
1.959
0
GA [5]
122.42
161.08
0.56
0.21
3
3
2.038
0
PSO [16]
106.69
155.89
0.897
0.28
2
2
2.272
0
HRDE [48]
–
–
–
–
–
–
2.0461
–
SA-PS [42]
–
–
–
–
–
–
2.313
–
AIA [48]
–
–
–
–
–
–
2.12
–
DERE [44]
–
–
–
–
–
–
2.046
–
ABC [44]
–
–
–
–
–
–
2.118
–
DE [44]
–
–
–
–
–
–
2.136
–
HABC [45]
–
–
–
–
–
–
2.046
–
HRTLBO [46]
–
–
–
–
–
–
2.046
–
ACO [14]
103.05
162.02
0.9
0.24
–
–
1.626
not considering Eq. (40)
FA [47]
98.4102
162.2882
0.82
0.2582
3
3
1.824
Eq. (24)
Note: “–” denotes that the authors do not provide the specific value in their works.
Comparison of the experimental results among different algorithms (when <inline-formula id="ieqn-214"><mml:math id="mml-ieqn-214"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, <inline-formula id="ieqn-215"><mml:math id="mml-ieqn-215"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:math></inline-formula> mm)
Algorithm
Vr(m/min)
Vs(m/min)
fr(mm/rev)
fs(mm/rev)
dr(mm)
ds(mm)
UC($/piece)
Constraint violation
GQBA (Ours)
119.1607
164.2276
0.6562
0.2624
2.6673
2.6652
2.4385
0
HRDE [48]
–
–
–
–
–
–
2.4791
–
DERE [44]
–
–
–
–
–
–
2.4793
–
HABC [45]
–
–
–
–
–
–
2.4790
–
AIA [48]
–
–
–
–
–
–
2.51
–
ABC [44]
–
–
–
–
–
–
2.503
–
DE [44]
–
–
–
–
–
–
2.512
–
SS [10]
–
–
–
–
–
–
2.5417
–
SA-PS [42]
–
–
–
–
–
–
2.7411
–
Note: “–” denotes that the authors do not provide the specific value in their works.
Comparison of the experimental results among different algorithms (when <inline-formula id="ieqn-223"><mml:math id="mml-ieqn-223"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>θ</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, <inline-formula id="ieqn-224"><mml:math id="mml-ieqn-224"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>6</mml:mn></mml:math></inline-formula> mm)
Algorithm
Vr(m/min)
Vs(m/min)
fr(mm/rev)
fs(mm/rev)
dr(mm)
ds(mm)
UC($/piece)
Constraint violation
GQBA (Ours)
109.6672
169.9682
0.5655
0.2261
3
3
2.0279
0
FPA [8]
109.6631
169.9785
0.5655
0.2262
3
3
2.0351
0
HPSO [7]
109.6655
169.9796
0.5655
0.2262
3
3
2.0351
0
COA [9]
117.9322
123.1993
0.5655
0.2262
3
3
2.2390
0
Optimal result obtained by the GQBA (when <inline-formula id="ieqn-231"><mml:math id="mml-ieqn-231"><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mi>θ</mml:mi><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>r</mml:mi></mml:mrow></mml:msub><mml:mo>+</mml:mo><mml:mo stretchy="false">(</mml:mo><mml:mn>1</mml:mn><mml:mo>−</mml:mo><mml:mi>θ</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>, <inline-formula id="ieqn-232"><mml:math id="mml-ieqn-232"><mml:msub><mml:mi>d</mml:mi><mml:mrow><mml:mi>t</mml:mi></mml:mrow></mml:msub><mml:mo>=</mml:mo><mml:mn>8</mml:mn></mml:math></inline-formula> mm)
Algorithm
Vr(m/min)
Vs(m/min)
fr(mm/rev)
fs(mm/rev)
dr(mm)
ds(mm)
UC($/piece)
Constraint violation
GQBA (Ours)
106. 0599
164.2371
0.6563
0.2624
2.6671
2.6656
2.5494
0
From Table 8, it can be clearly seen that the GQBA presented in this paper outperforms many algorithms, and its performance is comparable with that of some previously best approaches such as the COA [9], hybrid particle swarm optimization (HPSO) [43] and the FPA [8]. All four methods can produce an optimal result of ∼1.959. In contrast, the performance of the remaining methods is much worse than that of the GQBA; the corresponding production costs are larger than 2.0 or the resulting solutions are infeasible. In Table 9, the GQBA can obtain the minimum production cost among all the algorithms when the depth-of-cut is 8 mm. The proposed GQBA can further reduce the production cost obtained by hybrid robust differential evolution (HRDE) [48], the differential evolution algorithm with receptor editing (DERE) [44] and the hybrid ABC (HABC) [45], and it can obtain much better solutions than the other five comparative methods. In general, the GQBA can reduce costs by 2% to 12% compared to previously reported methods.
As mentioned previously, different definitions have been proposed for the tool life expression in previous studies [5,8,9], i.e., Tp=θTr+(1−θ)Ts. Because the formulation of tool life plays an important role in computing the production cost, the application effect of the GQBA in this case is also considered, as shown in Tables 10 and 11. From Table 10, it is obvious that the GQBA can obtain a minimum production cost of 2.0279, which outperforms all other methods. The optimal results and machining parameters are summarized in Table 11 with a depth-of-cut of 8 mm, which, as far as we know, has not been stated in previously published literature. In addition, this is the first study to use a BA variant to reduce the unit production cost for multi-pass turning operations, and our results show that the proposed GQBA can address the optimization problem efficiently and achieve better results than other methods.
Discussion
The GQBA can find better results than the other algorithms to further cut down the unit production cost. The main reasons come from the following two aspects.
Firstly, we improve the original BA algorithm to form the novel GQBA. The first improvement is to incorporate the current optimal positions of quantum bats and the global best position into the stochastic attractor to facilitate population diversification. The second improvement is to use a Gaussian distribution instead of the uniform distribution to update the positions of the quantum-behaved bats, thus performing a more accurate search and avoiding premature convergence. All these are verified by the numerical simulation experiments in Section 4.
Secondly, to overcome the complicated optimization problems in various fields, we need to carefully consider the characteristics of the specific problem and use the specific domain knowledge to design the optimization algorithm. In this paper, for the optimization problem of multi-pass turning, because the machining process can be divided into different numbers of roughing cuts, we decompose the whole optimization problem into several simple sub-problems according to the different numbers of roughing cuts. Each sub-problem can be conquered individually, which greatly reduces the space of the problem solution.
Therefore, the performance of the combination of traditional divide-and-conquer strategy and swarm intelligence algorithm is better than other algorithms that only use traditional mathematical methods or swarm intelligence algorithms. However. the proposed GQBA may be suitable to solve the optimization problem of multi-pass turning, but it is not general as other algorithms, that is, they can also be applied to the optimization problems in other manufacturing fields. Therefore, the generality of our algorithm is insufficient, and we hope to improve it further in the future.
Conclusions
In this research, the GQBA is developed to promote the efficiency of the BA and QBA with respect to accuracy and stability. In the GQBA, the combination of the QBA and a Gaussian distribution can expand the search space and prevent premature convergence. The modification of the stochastic attractor can contribute to promoting swarm diversity. The GQBA also inherits the characteristics of the QBA, such as simplicity and feasibility. To conclude, the experimental results of numerical functions verify the effectiveness of the GQBA. In addition, the results of multi-pass turning operation optimization show that the GQBA is a good alternative method.
For future research directions, the proposed GQBA can be utilized with the angle modulation technique [50–52] to solve binary optimization problems. Moreover, the GQBA can be applied to other real-world problems, including artificial neural networks, task scheduling, feature selection and image segmentation.
The data that supports the findings of this study are authentic and reliable. And the authors appreciated the reviewers for their helpful suggestions which greatly improved the presentation of this paper.
Funding Statement
This research was supported by the the National Natural Science Foundation of Fujian Province of China (2020J01697, 2020J01699).
Conflicts of Interest
The authors declare that they have no conflicts of interest to report regarding the present study.
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