Int J Adv Manuf Technol DOI 10.1007/s00170-007-1013-0
ORIGINAL ARTICLE
Multi-objective optimization of surface grinding process with the use of evolutionary algorithm with remembered Pareto set A. Slowik & J. Slowik
Received: 23 November 2006 / Accepted: 13 March 2007 # Springer-Verlag London Limited 2007
Abstract In this paper the multi-objective optimization of a surface grinding process making use of an evolutionary algorithm is presented. Such factors as wheel speed, workpiece speed, depth of dressing and lead of dressing are optimized in order to minimize production cost and surface roughness or to minimize production cost and maximize production rate. In the algorithm, the optimization is introduced in Pareto’s sense, all acceptable and nondominated solutions are remembered, and therefore the final result is not a single solution, but a whole set. The proposed method based on an example chosen from literature is tested, and the results obtained are compared with the results obtained by the use of other methods. Keywords Surface grinding . Evolutionary algorithm . Multi-objective optimization 1 Introduction The optimization of the surface grinding process is usually based on a simultaneous minimization of the production cost and a maximization of the production rate, or a simultaneous minimization of the production cost and a minimization of the
A. Slowik (*) Department of Electronics and Computer Science, Technical University of Koszalin, Sniadeckich 2 Street, 75-453 Koszalin, Poland e-mail:
[email protected] J. Slowik Department of Mechanical Engineering, Technical University of Koszalin, Raclawicka 15-17 Street, 75-620 Koszalin, Poland
grinding surface roughness. Because many mathematical dependencies describing the grinding process have recently been elaborated, a computer-aided optimization of this process is possible. Many studies describing methods of optimization of the grinding process have recently been created. Among them, we can mention [1], which concentrates mainly on the constraints of optimization. In [2] the technique of a simultaneous optimization of grinding and dressing parameters is described by the maintenance of the workpiece burn and the surface finish. Paper [3] presents quadratic programming for a multi-objective optimization of the grinding process parameters. In relation to a given problem, the genetic algorithms or ant colony optimization has also been used. In [4], the genetic algorithm used in an optimization of grinding process parameters with respect to a single criterion is presented. In [5], the genetic algorithm for a multi-objective optimization of grinding process parameters is described. However, in [6], an ant colony optimization is used to deal with the same problem. The methods described in [5, 6] are used for a multi-objective optimization of grinding process parameters with an objective function being the weighting sum of particular criteria values. The weighting sum method depends on finding a hyperplane tangential for a non-dominated solutions set. Its gradient is determined by the values of weight coefficients [7]. The solution of a multi-objective task is tangential points of this hyperplane with a set of nondominated solutions [7]. However, the weighting sum method used in [5, 6] causes some constraints. In the case when a set of acceptable solutions is concave then this method does not find polyoptimal solutions being in the concavity area of this set [8]. Besides, the solutions obtained are highly dependent on the assumed values of weight coefficients.
Int J Adv Manuf Technol
In this paper, a multi-objective optimization of the surface grinding process parameters using an evolutionary algorithm [7, 10, 11] is presented. In the algorithm, an objective function based on the domination level of particular solutions is used. Consequently, the result of the algorithm operation is not a single solution dependent on the assumed values of weight coefficients (as in [5, 6]), but it is a whole set (also called Pareto front or Pareto optimal solutions set). It is worth to point out that in the case when a set of acceptable solutions is concave, the proposed approach makes it possible to find polyoptimal solutions occurring in the concavity area of this set. The proposed method is called MOEA-SGP (Multi Objective Evolutionary Algorithm for Surface Grinding Problem), and it is tested on an example chosen from literature [3, 5, 6]. The results obtained with the use of the method described are compared with the results obtained with the use of other methods: quadratic programming (QP) [9], genetic algorithm (GA) [5] and ant colony optimization (ACO) [6].
h iT is the vector of decision where x ¼ x1* ; x2* ; . . . ; xn* variables. 2.3 Definition of Pareto optimality A point x*∈Ω is Pareto optimal if for every x∈Ω and I= {1, 2, ..., k} either, 8i2I fi x* fi ðxÞ ð5Þ and, there is at least one i ∈ I such that f i x* < f i ð xÞ
ð6Þ
In words, this definition says that x* is Pareto optimal if there exists no feasible vector x that would decrease some criterion without causing a simultaneous increase in at least one other criterion. The phrase “Pareto optimal” means with respect to the entire decision variable space unless otherwise specified [13]. 2.4 Definition of Pareto dominance
2 Basic concepts In this point the definitions of global minimum, general multi-objective optimization problem (MOP), Pareto optimality, Pareto dominance, and Pareto optimal set, are presented [13]. The bold symbols represent a vector.
n Given a function f : Ω R ! R; Ω 6¼ 0 for x 2 Ω the Δ * * value f ¼f x > 1 is called a global minimum if and only if: 8x 2 Ω : f x* f ðxÞ ð1Þ
Then, x* is the global minimum solution, f is the objective function, and the set Ω is the feasible region (Ω∈S), where S represents the whole search space. 2.2 Definition of general multi-objective optimization problem (MOP) h iT Find the vector x ¼ x1* ; x2* ; . . . ; xn* which will satisfy the m inequality constraints: i ¼ 1; 2; . . . ; m
ð2Þ
the p equality constraints hi ð xÞ ¼ 0
i ¼ 1; 2; . . . ; p
ð3Þ
and will optimize the vector function f ðxÞ ¼ ½f1 ðxÞ; f2 ðxÞ; . . . ; fk ðxÞT
2.5 Definition of Pareto optimal set For a given MOP f(x), the Pareto optimal set (P*) is defined as:
2.1 Definition of global minimum
gi ð xÞ 0
A vector u=(u1,...,uk) is said to dominate vector v = (v1,..., vk) (denoted by uv) if and only if u is partially less than v, i.e., ∀i∈{1,...,k},ui≤vi⋀∃i∈{1,...,k}: ui U * otherwise
WRP WRP G ; when WWP Ra* E4 ¼ 0; otherwise
ð31Þ
solution better than it considering P all thecriteria at the same M time. A non-dominated and permissible i¼1 di ¼ 0 individual (which does not disturb any constraint Err=0) has the value of objective function FC=1. In the third step, the evolutionary algorithm termination condition is checked. The number of generations resulting from the algorithm convergence is accepted as its terminate criterion. In the case of a termination of the algorithm, step ninth is executed, otherwise step four is executed. In the fourth step, all acceptable and non-dominated solutions, (Pareto optimal solutions) for which the objective function FC values one in the current generation, are remembered in Popt set. In the fifth step, the updating of acceptable and nondominated solutions set Popt is accomplished. This actualization is based on a look-through a whole Popt set, and removing the dominated solution from it. In the sixth step, a selection of individuals using a M roulette method [10] is accomplished. Next, the 10 randomly chosen individuals from Popt set are introduced to random places in the newly created population. In the seventh step, the crossover and mutation operators are used in the population. The crossover operator is based on cutting of two randomly chosen (with probability pcross) individuals from the population in a randomly chosen cut point, and exchanging the cut fragments of its
E5 ¼ 0
ð32Þ
Table 2 Results obtained for finish grinding; W1 =0.3, W2 =0, W3 = 0.7 (aw =0.055 [mm], ap =0.0105 [mm/pass], WRP≥ 20[mm3/min-N])
–
for CT minimization, and Ra minimization (finish grinding)
di ¼
1; 0;
when ðCT < CT i Þ ^ ðRa < Rai Þ otherwise
E4 ¼ 0
ð28Þ
( WRP WRP* ; E5 ¼ 0; –
ð27Þ
when WRP < WRP* otherwise
ð29Þ
for CT minimization, and WRP maximization (rough grinding) (
di ¼
1; when ðCT < CT i Þ ^ ðWRP > WRPi Þ 0; otherwise:
ð30Þ
i - number of individual, M - size of population in an evolutionary algorithm. The remaining marks are identical as in section 4. The value of objective function FC for every individual is the greater the larger number of solutions from the population dominate the individual, and the greater number of constraints it disturbs. A dominated solution means a solution for which there is in the population at least one
Vs Vw doc L CT WRP Ra COF
QP
GA
ACO
MO
2000 19.99 0.052 0.091 7.7 20.00 0.83 0.554
1986 21.40 0.024 0.136 6.6 20.08 0.83 0.521
2023 19.36 0.019 0.134 6.9 20.00 0.76 0.505
2022 21.95 0.013 0.136 6.5 20.02 0.79 0.504
Int J Adv Manuf Technol Table 3 Ten randomly chosen solutions from Popt set for finish grinding No.
Vs
Vw
doc
L
CT
WRP
Ra
1 2 3 4 5 6 7 8 9 10
2023 2023 2023 2023 2023 2023 2023 2023 2023 2023
10.08 10.66 11.90 13.17 14.61 15.64 16.91 18.78 21.46 22.69
0.111 0.102 0.089 0.075 0.063 0.055 0.045 0.034 0.022 0.012
0.137 0.137 0.137 0.137 0.137 0.137 0.137 0.137 0.137 0.137
7.5 7.3 6.9 6.6 6.3 6.1 5.9 5.6 5.3 5.2
25.71 25.24 24.54 23.82 23.12 22.69 22.14 21.49 20.83 20.11
1.79 1.79 1.80 1.80 1.80 1.80 1.79 1.79 1.80 1.76
chromosomes between them. In Fig. 5, two example individuals A and B chosen randomly to crossover in a randomly chosen cut point are shown. In Fig. 6, child individuals A’ and B’ created by the crossover operation from Fig. 5 are presented. However, mutation depends on the drawing for a new value for a randomly chosen (with probability pmut) gene in the population. An example chromosome A with gene randomly chosen to mutation is shown in Fig. 7. In Fig. 8, the child chromosome A’ created by the mutation operation from Fig. 7 is presented. In the eighth step, we return to the second step. In the ninth step, the solutions remembered in Popt set are derived and the algorithm (method MOEA-SGP) is stopped. A set of acceptable (fulfilling required constraints) and non-dominated solutions is the operation result of MOEASGP method. Moreover, the choice of one single solution from whole set Popt, as a final solution, is an open question. In this paper (for the purpose of the results comparability from papers [5, 6]) as a chosen criterion of a single solution from set Popt, the minimal value of parameter COF (described in the next section of this work by formula (33)) is accepted.
7 Description of experiments The following input data (identically as in papers [5, 6]) for the purpose of conducting experiments are accepted: – – – – – – – – – – – – – – – – – –
No. of work piece on table p (pc)=1 Length of the work piece (mm) Lw =300 Empty length of grinding (mm) Le =150 Width of work piece (mm) bw =60 Empty width of grinding (mm) be =25 Cross feed rate (mm/pass) fb =2 Total thickness of cut (mm) aw =0.1 Down feed of grinding (mm/pass) ap =0.0505 Number of spark out grinding (pass) Sp =2 Expected production cost limitation ($/pc) CT*=9.8833 Diameter of wheel (mm) De =355 Width of wheel (mm) bs =25 Grinding ratio G=60 Distance of wheel idling (mm) Sd =100 Speed of wheel idling (mm/min) Vr =254 Time of loading and unloading work piece (min) ti =5 Time of adjusting machine tool (min) tch =30 Total number of workpiece to be ground between two dressing (pc) Nd =20
Table 4 Ten randomly chosen solutions from Popt set for rough grinding No.
Vs
Vw
doc
L
CT
WRP
Ra
1 2 3 4 5 6 7 8 9 10
2022 2022 2022 2023 2023 2023 2023 2023 2023 2023
10.14 11.45 11.92 12.67 14.57 15.61 17.18 19.03 21.00 21.97
0.041 0.036 0.035 0.032 0.029 0.025 0.022 0.018 0.015 0.014
0.136 0.137 0.136 0.137 0.135 0.136 0.136 0.137 0.137 0.136
9.9 9.2 9.0 8.6 8.0 7.7 7.3 6.9 6.6 6.5
20.00 20.05 20.08 20.03 20.02 20.00 20.05 20.05 20.06 20.08
0.64 0.66 0.67 0.68 0.70 0.72 0.74 0.76 0.78 0.79
Int J Adv Manuf Technol Fig. 11 Number of solutions in Popt set at function of crossover and mutation probability for finish grinding
– – – – – – – – – –
Batch size of workpiece Nt =12 Total no. of workpiece to be ground during the life of dresser (pc) Ntd =2,000 Cost of dresser ($) cd =25 Cost of wheel per mm3 (cost/mm3) cs =0.003 Workpiece hardness (Rockwell hardness) Rc =58 Surface finish limitation-rough (m) Ra*=1.8 Cost per hour labor and administration ($/h) Mc =26.84 Wheel bond percentage (%) VOL=6.99 Grain size (mm) dg =0.3 Workpiece removal parameter limitation (mm3/min-N) WRP*=20
Fig. 12 Value of COF coefficient at function of crossover and mutation probability for finish grinding
– – – – –
Static machine stiffness (N/mm) Km =100,000 Dynamic machine characteristics Rem =1 Initial sear flat area percentage Ao =0 Wear constant (mm−1) ku =3.937·10−7 Constant dependent on coolant and grain type Ka = 0.0869
The following range of optimization variables (as in papers [5, 6]) is accepted: – – – –
Vs (m/min): MIN Vs =1000; MAX Vs =2023 Vw (m/min): MIN Vw =10; MAX Vw =22.7 doc (mm): MIN doc=0.01; MAX doc=0.137 L (mm/rev) MIN L=0.01; MAX L=0.137
Int J Adv Manuf Technol Fig. 13 Number of solutions in Popt set at function of crossover and mutation probability for rough grinding
The following parameters for evolutionary algorithm are accepted: – – – –
Probability Probability Number of Number of
of crossover (pcross)=0.1 of mutation (pmut)=0.1 individuals in population (M)=100 generations=3,000
Additionally (as in papers [5, 6]) it is assumed that during Ra minimization and CT minimization (finish grinding): aw =0.055 [mm], ap =0.0105 [mm/pass]. Howev-
Fig. 14 Value of COF coefficient at function of crossover and mutation probability for rough grinding
er, during WRP maximization and CT minimization (rough grinding): aw =0.1 [mm], ap =0.050 [mm/pass]. In Fig. 9, solutions for finish grinding obtained with the use of MOEA-SGP method are presented. In finish grinding, the surface roughness (Ra), and production cost (CT) are minimized. After the algorithm termination, a set of Pareto optimal solutions (Popt) composed of 239 accepted and non-dominated solutions is obtained. In Fig. 10, the solutions for rough grinding obtained with the use of MOEA-SGP method are presented. In rough
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grinding, the production rate (WRP) is maximized, and the production cost (CT) is minimized. After the algorithm termination, a set of Pareto optimal solutions (Popt) composed of 145 accepted and non-dominated solutions is obtained. We can see from Figs. 9 and 10 that while using MOEASGP method, it is possible to obtain many equivalent solutions for a given problem. An open question is the choice of a single solution from this set. In order to compare results obtained with the use of MOEA-SGP method with the results obtained in papers [5, 6], the value of coefficient COF is computed for each individual from Pareto optimal set Popt (identically as in papers [5, 6]) according to the following formula:
COF ðVs ; Vw ; doc; LÞ ¼ W1
times, and the average value of the results obtained was accepted as the output value (the algorithm was stopped every time after the achievement of 1,000 iterations). It was investigated in what manner the value of COF parameter and the number of individuals in Popt set are dependent from the value of the crossover and mutation probability. The results obtained in a graphical form were presented in Figs. 11 and 12 (finish grinding), as well as in Figs. 13 and 14 (rough grinding). It can be seen from Figs. 11, 12, 13, and 14 that the results obtained are to a larger extent sensitive to the change of mutation probability than to the change of crossover probability. The best results are achieved for mutation probability in the range [0.1; 0.4]. The crossover parameter value can be altered in the range from [0.1 to 1], without any substantial changes of the results obtained.
CT WRP Ra þ W2 þ W3 CT * WRP* Ra* ð33Þ
Next, the minimal value of coefficient COF is accepted as a choice criterion of a single solution from Pareto optimal set Popt (obtained with the use of MOEA-SGP method, and presented in Figs. 9 and 10). In Table 1 (rough grinding), and in Table 2 (finish grinding) the results obtained with the use of MOEA-SGP method in comparison to the results obtained in papers [5, 6] are presented. The marks in tables are as follows: QP - quadratic programming, GA - genetic algorithm, ACO - ant colony optimization, MO - result obtained with the use of MOEASGP method described. The result of MOEA-SGP method operation is not a single solution, but the whole family of solutions remembered in Popt set. Thanks to this, we obtained 239 equally important solutions for finish grinding, and 145 solutions for rough grinding. Each of them is a non-dominated and acceptable solution. In Table 3, ten randomly chosen solutions from Popt set (see Fig. 9) for finish grinding are shown. In Table 4, ten randomly chosen solutions from Popt set (see Fig. 10) for rough grinding are shown. The solutions shown in Tables 3, and 4, satisfy all the constraints and are non-dominated.
8 Sensitivity analysis of MOEA-SGP method In order to conduct a sensitivity analysis of the algorithm on its input parameters, the size of population was accepted to be 100, while the values of the probability of crossover and mutation were changed from 0.1 to 1 with a step every 0.1. For individual settings, the algorithms were started ten
9 Conclusion From Table 1, it is observed that for rough grinding, the production cost (CT) obtained with the use of MOEA-SGP method is 14% higher than QP, 1.4% higher than GA and 4% lower than ACO, but the work-piece removal (WRP) for MOEA-SGP is 31% higher than QP, 14% higher than GA and 4% higher than ACO. In the case of the finish grinding (Table 2) it is observed that production cost obtained by MOEA-SGP is 16% lower than QP, 1.5% lower than GA and 6% lower than ACO, but the surface finish for MOEA-SGP is 4.8% lower than QP, 4.8% lower than GA, and 3.8% higher than ACO. It is necessary to point out that the result of MOEA-SGP method operation is not a single solution, but the whole family of solutions remembered in Popt set. Each of them is equally important and the choice of only one solution from this set is an open question (see Tables 3 and 4). When the minimal value of formula (33) is assumed as a choice criterion, as seen in Tables 1 and 2, in all cases better values of COF parameter are obtained with the use of MOEA-SGP method. MOEA-SGP method has achieved an overall improvement of 52% over QP, 29% over GA and 13% over ACO in rough grinding (Table 1) and 9% over QP, 3.3% over GA and 0.2% over ACO in finish grinding operations when considering overall objective criteria (using formula (33) identically as in paper [5, 6]). Better results obtained the use of MOEA-SGP method in relation to other methods could suggest that this method has a better exploration possibility of potential solutions space. It is necessary to point out that MOEA-SGP method does not possess any constraints connected with the shape of solutions set. Owing to operating on dominance relations between individuals it is possible to search solutions in the concavities area of these sets.
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