SIMULATED ANNEALING ASSISTED OPTIMIZATION OF FUZZY RULES FOR MAXIMIZING TOOL LIFE IN HIGH-SPEED MILLING PROCESS Asif Iqbal CMEE, Nanjing University of Aeronautics & Astronautics. 29-Yu Dao Street, Nanjing 210016, P.R. China
[email protected]
Ning He CMEE, Nanjing University of Aeronautics & Astronautics. 29-Yu Dao Street, Nanjing 210016, P.R. China
[email protected]
Liang Li CMEE, Nanjing University of Aeronautics & Astronautics. 29-Yu Dao Street, Nanjing 210016, P.R. China
[email protected]
The shortened tool life becomes more of the serious issue when HSM is applied to the machining of steels in their hardened state. Elevated hardness values of heat-treated steels (up to 63HRc) have been proved to be significant in reducing the tool life. Lot of experiment-based research has been carried to find out the ways for enhancement of tool life, most of which focuses upon investigation of effects of cutting and tooling parameters. References [2 – 8] list some of the papers that are noteworthy in this regard and recommend values for different parameters with respect to given machining conditions. Study of these papers and other literature leads to a general conclusion that the tool life can be increased by decreasing the cutting speed, feed rate, and depths of cut (axial and radial). Poulachon et al. [9] described that for high-speed cutting of hardened steels, the hierarchy of influential parameters upon tool life is: workpiece material hardness, cutting speed, feed rate, and depth of cut. Decrease in values of the aforementioned parameters may lead to increase in tool life, but on the other hand, it also results in the reduction of material removal rate (MRR) and the reduced MRR means loss of productivity and, ultimately, increase in production cost. In order to cope with this two sided problem, a common practice is taken by the industry and that is to optimize the cutting parameters while fixing the minimum value of MRR for given machining conditions. Axinte et al. [8] reported that for hardened steels, cutting speeds are generally in range of 200-400m/min, feed rates 0.05-0.2mm/tooth, and axial and radial depths of cut 0.2-2mm. In this paper an expert system based approach has been presented for optimization of the cutting speed for the given combinations of workpiece material hardness, feed rate, and axial and radial depths of cut, and for the fixed minimum value of MRR. Fuzzy logic is used as the reasoning mechanism of the rule base in order to cope with uncertainties and vagueness of input, output, and processing. In order to take the rule base to its maximum effectiveness, in terms of accuracy of predicting the output values, optimization of rules is sought. Simulated
ABSTRACT In metal cutting industry it is a common practice to search for optimal combination of cutting parameters in order to maximize the tool life for a fixed minimum value of material removal rate (MRR). After the advent of highspeed milling (HSM) process, lot of experimental research has been done for optimization of the parameters. It is highly beneficial to convert raw data into a comprehensive knowledge-based expert system using fuzzy logic as the reasoning mechanism. In this paper an attempt has been presented for the optimization of the rules so as to have the effective most knowledge-base for given set of data. Experiments were conducted to determine the best values of cutting speeds that can maximize tool life for different combinations of input parameters. Triangular fuzzy sets were developed for input parameters and the cutting speed, based upon the range of values obtained. Max-min strategy was used for aggregation of fuzzy rules. Simulated annealing algorithm was used to work out the optimal combination of fuzzy rules out of 5.815 × 1025 possible combinations. Optimized combination of fuzzy rules provided the estimation error of only 7.18m/min as compared to 232m/min of that of randomized combination of rules. KEY WORDS Simulated annealing, fuzzy logic, expert system, HSM
1. Introduction High-speed milling (HSM) has confirmed its place in manufacturing industry as a well-established production technology owing to numerous advantages like reduction of cutting forces, removal of process heat along with the chip, improved dimensional accuracy, better workpiece surface finish, and machining in the range not subject to critical vibrations. Besides these advantages, HSM also comes with a serious drawback, which is the unacceptable shortening of tool life [1].
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Naeem Ullah Dar Department of Mechanical Engineering, University of Engineering & Technology, Taxila, MED, UET Taxila, Pakistan
[email protected]
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Tool life per unit axial depth of cut (mm2)
annealing algorithm has been utilized for optimization of the fuzzy rules.
2. Experimental Work The pertinent data were taken from the series of highspeed rough end milling experiments. Experiments were performed in order to determine the values of cutting speed (Vc) – for different combinations of workpiece material hardness, axial depth of cut (ap), and radial depth of cut (ae) – that result in maximum tool life provided MRR should not fall below a specific value. In order to reduce the number of experiments and the complexity of the model, the feed per tooth (fz) was fixed to 0.1mm for all experiments. Survey of literature [2-8] also suggests that feed rate of 0.1mm/tooth has been used in most of the cases. Two kinds of hardened steels were used in the experiments: (1) AISI 4340 and (2) AISI D2. Table 1 shows the different levels of parameters tested in the experiments. Commonly the MRR values employed in industry for high-speed rough end milling of hardened steels vary from 2000 to 2600mm3/min. In this case, the minimum value of MRR was fixed to 2300mm3/min. The mathematical expression for MRR, in terms of the cutting parameters, can be represented as follows: MRR
ae a p zf zVc
D
25000
AISI 4340 (45HRc)
20000
AISI 4340 (50HRc) AISI D2 (55HRc)
15000 10000 5000 0 50
150
250
350
Optimal cutting speed Vc* (m/min)
Fig. 1. Relationship between Optimal Cutting Speed and Corresponding Tool Life per Unit ADoC Figure 1 shows the relationship between experimentally determined optimal values of cutting speed, for the three workpieces, and the corresponding values of tool life per unit axial depth of cut (ADoC) obtained. All the points provided MRR higher than the fixed value. For each level of hardness, it can be observed that the increase in optimal cutting speed results in decreased tool life. It can also be observed that, on average, for any given value of optimal cutting speed, the workpiece material of higher hardness results in decreased tool life. 2.1 The Model Nonlinear regression was used to develop the empirical model from the data obtained from experiments. The squared multiple correlation factor for the model obtained is 0.9708, which means that the model accounts for 97.1% of the variability in the obtained data. The model for evaluating optimal cutting speed is as follows:
(1)
In above expression, z represents the number of cutting edges and D represents the diameter of tool (end mill). For all the experiments, the cutting tools used were flat end solid K30 carbide cutters with PVD coated mono layer of TiAlN, having diameter (D) of 8mm, corner radius (R) of 1.5mm, rake angle (γ) of 5º, flank angle (α) of 6º, and number of flutes (z) equal to 4. The experiments were performed on Micron UCP 710, 5-axis, vertical milling center. No coolant was used in any experiment and down-milling was utilized as the milling orientation. Tool failure criteria employed was the maximum flank wear of 0.2μm. The flank wear was measured using tool maker’s microscope. Table 1 suggests 33 = 27 possible combinations of 3 levels of 3 factors involved. For each combination, experiments were conducted in order to find the value of speed that can provide maximum tool life and the MRR values of more than 2200mm3/min. Detailed analysis of experimental results obtained is out of scope of this paper, but the data were converted into empirical model using nonlinear regression.
Vc* = 2366.084 – 0.0000000228(hardness)4.786 – 206.937(ae)1.2347 – 1834.164(ap)0.1022 (2) For instance, for milling of workpiece having material hardness of 45HRc at radial and axial depths of cut of 0.6mm and 2mm respectively the model recommends the optimal cutting speed of 285m/min giving the MRR of 2722mm3/min. On the other hand the actual experimental results showed that for these values of parameters, the optimal cutting speed is 276m/min, providing tool life of 9042mm2 of material removed per unit axial depth of cut.
3. The Fuzzy Expert System The empirical model (equation 2) contains some degree of impreciseness. This impreciseness is the mix of modeling impreciseness as well as the process impreciseness. It is quite rare that two, exactly similar, tools used in milling of same workpiece material using exactly similar cutting parameters, end up with exactly equal tool life. This kind of variability and impreciseness can be well handled using fuzzy logic. Fuzzy sets and logic is a discipline that has proved itself successful in automated reasoning of expert systems [10].
Table 1. Levels of Parameters Tested in the Experiments. fz was fixed to 0.1mm/tooth S/No Hardness (HRc) ae (mm) ap (mm) 1 45 (AISI 4340) 0.6 2 2 50 (AISI 4340) 0.7 3 3 55 (AISI D2) 0.8 5
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Index 0 1 2
nine cutting speed states that are to be determined. Therefore, maximum number of rules possible are 27 (= 3 × 3 × 3). An important question arises here, “which cutting speed states to be assigned to 27 possible combinations of input fuzzy sets”? For a simple 2-inputs 1-output fuzzy model, the designer has to select the most optimum set of fuzzy rules from more than 10,000 combinations [12]. For this study, there are 27 fuzzy rules with 9 possibilities each. Thus the total number of possible fuzzy rules combination will be 927 = 5.815 × 1025. For the most optimal possible combination of rules, simulated annealing algorithm has been employed in this study and explained in section 4.
Table 2. Fuzzy Expressions for Hardness Abbreviation Expression LH Less Hard MH Medium Hard HH Highly Hard
Table 3. Fuzzy Expressions for Radial Depth of Cut (RDoC) Index Abbreviation Expression 0 L Light 1 N Normal 2 H Heavy Table 4. Fuzzy Expressions for Axial Depth of Cut (ADoC) Index Abbreviation Expression 0 S Shallow 1 M Medium 2 D Deep
The program used to optimize the fuzzy rules involves very important routines of input data fuzzification and output data defuzzification. Different strategies produce different results. There are two methods commonly used to yield the aggregation of rules. They are max-min inference method and max-product method. The detail of both methods can be read from [13]. For the current research work, the max-min strategy has been utilized for yielding aggregation of the fuzzy rules.
Table 5. Fuzzy Expressions for Cutting Speed Abbreviation Expression ES Extremely slow VS Very slow SL Slow SS Slightly slow ME Medium SF Slightly fast FA Fast VF Very fast EF Extremely fast
1
Membership function
Index 0 1 2 3 4 5 6 7 8
3.3 Aggregation of Fuzzy Rules
LH
MH
HH
0
3.1 Fuzzy Sets
45
47
49
51
53
55
57
59
Hardness (HRc)
Equally distributed triangular shaped fuzzy sets are employed for three input variables: hardness, radial depth of cut, and axial depth of cut; as well as for output variable: cutting speed. Table 2, 3, and 4 show the fuzzy expressions used for input variables and table 5 shows those for output variable. Membership function for each fuzzy set for input variables are shown in figures 2, 3, and 4, and those for output variable are shown in figure 5. Number of fuzzy sets and corresponding range for each variable are selected based upon range of data of input and output variables. Based upon these selections, we are supposed to optimize the fuzzy rules. Optimization of fuzzy sets for input and output variables is out of scope of this paper and can be considered as topic for future research.
Fig. 2. Hardness Membership Function
Membership function
1
L
N
H
0 0.5
0.55
0.6
0.65
0.7
0.75
0.8
Radial depth of cut (mm)
Fig. 3. Membership Function for Radial Depth of Cut (RDoC)
Membership function
1
3.2 Fuzzy Rules The relationship between inputs and output in a fuzzy system is characterized by set of linguistic statements which are called fuzzy rules [11]. The number of fuzzy rules in a fuzzy system is related to the number of fuzzy sets for each input variable. In this study, there are three fuzzy sets for each of three input variables, and there are
S
M
D
0 1
1.5
2
2.5
3
3.5
4
4.5
5
Axial depth of cut (mm)
Fig. 4. Membership Function for Axial Depth of Cut (ADoC)
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Membership function
1
ES
VS
SL
SS
ME
SF
200
250
300
FA
VF
EF
0 0
50
100
150
350
400
450
500
Cutting speed (m/min)
Fig. 5. Cutting Speed Membership Function
4. The Optimization Table 7. The Optimal Combination of 27 Fuzzy Rules Rule Antecedents Consequent No. Hardness RDoC ADoC Speed 1 LH L S EF 2 LH L M SS 3 LH L D VS 4 LH N S FA 5 LH N M SS 6 LH N D VS 7 LH H S SF 8 LH H M SL 9 LH H D VS 10 MH L S FA 11 MH L M SS 12 MH L D SL 13 MH N S FA 14 MH N M SS 15 MH N D VS 16 MH H S SF 17 MH H M SL 18 MH H D VS 19 HH L S FA 20 HH L M SS 21 HH L D VS 22 HH N S FA 23 HH N M SS 24 HH N D VS 25 HH H S SF 26 HH H M SL 27 HH H D ES
Simulated annealing (SA) is a stochastic neighborhood search method, which is developed for combinatorial optimization problems [14]. It is based on the analogy between the process of annealing of solids and solution methodology of combinatorial optimization problems. The pseudo-code of the algorithm developed for optimization of fuzzy rules using SA technique, has been provided in the appendix to the paper. C++ was used to code the algorithm and to develop the sub-routine for fuzzification and defuzzification of the data involved. The SA parameters were operated using following values: (1) starting annealing temperature (T0) = 500m/min; (2) rate of cooling (α) = 0.98; (3) maximum number of iterations (imax) = 100; (4) length of Markov Chain at each iteration (L) = 5 × 9 × 9 = 405; (5) minimum acceptance ratio (Rf) = 0.01; (6) minimum number of accepted transitions at each iteration (ATmin) = 100. The objective function of the “optimization of fuzzy rules” problem is the minimization of estimation error, where the term “estimation error” can be defined as follows:
estimation _ error
l m n 1 est * (Vc Vc ) (3) l m n i 1 j 1 k 1
For equation 3:
2.
l, m, n = number of levels provided by the user for each of three variables: hardness, radial depth of cut, and axial depth of cut, respectively. Vc* = optimal cutting speed evaluated using equation
Estimated error (m/min)
Vcest = optimal cutting speed evaluated using given combination of fuzzy rules. Table 6. Levels of 3 Input Variables Used in the Program for Evaluation of Estimation Error Level Hardness Radial DoC Axial DoC 1 45HRc 0.6mm 2mm 2 50HRc 0.7mm 3mm
250 200 150 100 50 0 0
10
20
30
40
Iteration Number
Fig. 6. Variation of Estimated Error along Number of Iterations
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4.1 Optimization Results
Appendix
For the present case of optimization, the initial combination of fuzzy rules was made at random and the termination criteria for algorithm consisted of fulfillment of one of three conditions that has been provided in the pseudo-code of algorithm (see appendix). In order to determine the optimal combination of fuzzy rules, each transition of the rules for all of the iterations were tested for estimation error, using eight combinations of three variables with two levels each. Table 6 shows the detail. The program based upon SA algorithm kept processing for 31 iterations, until the termination criterion of “estimation error value did not improve for last 20 iterations” was fulfilled. At the termination of program, the optimal combination of fuzzy rules was printed out, which has been listed in table 7. For this optimal combination of rules, the testing values of input variables resulted in least value of estimation error, which is 7.18m/min. Figure 6 shows the continuous improvement in estimation error through the iterations of this program run. Considering the range of cutting speed (50 – 500m/min) being tackled by the fuzzy rule base and the uncertainties involved in the problem of improving tool life in HSM process, the error of 7.18 m/min can be considered as a minute one. The program was re-run using different values of input variables resulting in same optimal combination of fuzzy rules but with slightly different values of estimation error.
Following is the pseudo-code of simulated annealing algorithm for optimization of fuzzy rules: [0] Initialize [0.1] Set annealing parameters T0, ATmin, imax, α, Rf [0.2] Initialize iteration counter, i = 0 [0.3] Generate initial rules combination and calculate estimation error value, i.e. rules [0], error [0] [1] Execute outer loop, i.e. steps 1.1 to 1.7 until conditions in step 1.7 are met. [1.1] Initialize inner loop counter l = 0, and accepted number of transitions AT = 0 [1.2] Initialize rules combination for inner loop; rules [i][0] = rules [i] and error [i][0] = error [i] [1.3] Execute inner loop, i.e. steps 1.3.1 to 1.3.5 until conditions in step 1.3.5 are met [1.3.1] Update l = l + 1 [1.3.2] Generate a neighboring solution by changing randomly one rule, and compute estimation error for new rules combination (rules [i][l] and error [i][l]) [1.3.3] Assign q = error [i][l] – error [i][l – 1] [1.3.4] If q ≤ 0 or Random (0, 1) ≤ e-q/To then Accept rules [i][l] and error [i][l] Update AT = AT + 1 Else reject generated combination: rules [i][l] = rules [i][l – 1], error [i][l] = error [i][l – 1] [1.3.5] If one of following conditions hold true: AT ≥ ATmin; OR l ≥ 5S2 (S – No. of fuzzy sets of output variable), then assign length of Markov chain L [i] = l. Terminate inner loop and go to 1.4, else continue the inner loop and go to 1.3.1 [1.4] Update i = i + 1 [1.5] Update: rules [i] = rules [i – 1][L[i] – 1] and error [i – 1][L[i] – 1] [1.6] Reduce cooling temperature: T [i] = α.T[i – 1] [1.7] If one of following conditions hold true: i ≥ imax; OR (AT / L[i]) ≤ Rf; OR estimation error value does not reduce for last 20 iterations, then terminate the outer loop and go to 2, else continue outer loop and go to 1.1 [2] Print out the best rules combination along with minimum estimation error value and terminate the procedure
5. Conclusion In the presented paper, a program based upon simulated annealing algorithm was developed for optimization of fuzzy rule base that should precisely estimate the values of cutting speed that will maximize the tool life for given high-speed milling conditions. Following conclusions can be drawn from the discussion: 1. 2.
3.
Optimal cutting speed is a highly vague quantity and its determination process should involve treatment of fuzzy logic. For the development of fuzzy expert system, designer has to develop fuzzy rules from unmanageably highly number of options. Stochastic neighborhood search method, like simulated annealing, can very effectively provide the best combination of the rules. In the presented work, fuzzy rule base, for estimation of optimal cutting speed, was optimized using SA algorithm and utilization of max-min strategy of rules aggregation. The optimal rule base provided the output values with estimation error of 7.18m/min, as compared to 232m/min of that of randomized combination of rules.
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