Flank wear modeling of a tungsten carbide tip using the GMDH method in turning operation/Volframo karbido ploksteles uzpakalinio pavirsiaus issidevejimo tekinant modeliavimas grupinio duomenu apdorojimo metodu.

Ali, D. Kara ; Ghernaout, M.E.A. ; Galiz, S. 等

1. Introduction

Although machining or material removal with cutting tool processes is an oldest technique of manufacturing machine components, many experimental research works have been performed in this field during the last century. It was the improving tool life and machined surface quality, machine energy consumption, dynamic phenomena, etc ...

As a result, any further improvement of the machine, tool and process design must be justified through a series of experimental studies. However, any machining test includes a great number of independent variables and the traditional techniques of the Design of Experiments (DOE), becomes expensive and time consuming. Moreover, uncertainty of many included variables might affect the test outcome in metal cutting testing. To overcome these drawbacks discussed by Astakhov and Galitsky [1] concerning the traditional DOE, this paper uses the powerful method called Group Method of Data Handling (GMDH) as a process modeling tool for forming a statistical model of a complex multi-variables systems using a few process data. This method is introduced by the Ukrainian cyberneticist and engineer A.G. Ivakhnenko [2] for solving modeling and classification problems using the polynomial theory of complex systems. The Ivakhnenko authors [3] have published a review of problems solvable by algorithms of the GMDH. The objectiveness of GMDH algorithm and its satisfactory performance as a non-linear modeling approach has driven a number of researchers to investigate it further as well as test it in a broad spectrum of applications as data mining and knowledge discovery, forecasting and systems modeling, optimization and pattern recognition. The application of the GMDH method to manufacturing starts at the early of 1980s and different versions are considered: basic, modified and enhanced GMDH-type network. Several research works have been performed to build a mathematical model describing tool wear or tool life in the cutting process. Briefly, we indicate as a sample the following works performed since 1980 and found in the literature. Hence, Nagasaka and Hashimoto [4] have estimated the quality of chip disposal and Yoshida and al. [5] have identified the grinding wheel wear of the abrasive cut-off. The work published by Nagasaka and al. [6] has determined an optimum of combination of operating parameters in abrasive cut-off. The prediction and detection of the cutting tool failure has concerned the work performed by Uematsu and Mohri [7]. Jiaa and Dornfeld [8] have published a work concerning the prediction and detection of tool wear. El-Khabeery and El-Axir [9] have studied the effects of milling roller-burnishing parameters on surface integrity. The work performed by Astakhov and Galitsky [1] deals with the test of the tool life in gun-drilling and finally the work presented by Onwubolo and al. [10] concerns the modeling of tool wear in end-milling. Inductive GMDH algorithms are used because they provide a possibility to find automatically interrelations in data, to select the 'optimal' structure of model and to increase the accuracy of existing algorithms. This original self-organizing approach is substantially different from deductive methods commonly used in traditional DOE. This paper describes the use of the GMDH method to build a model for predicting the tool flank wear in a turning operation without lubrication. The aim is to analyze the influence of input variables on the tool life and serving promoting automation of cutting process.

2. Background on GMDH method

The GMDH method gives a procedure for modeling complex nonlinear systems from input and output data, based on the principle of heuristic self-organization. This method allows the discovery of the complex relationship between input and output variables objectively without having detail knowledge of the system investigated and a large number of data.

Suppose that the system equation for input variables [x.sub.i] and output y is y = f([x.sub.l],[x.sub.2],...,[x.sub.n]), where y denotes the complete description of the system. This function can be expressed by the Volterra series, discrete analogue of which is the Kolmogorov-Gabor polynomial

y = [b.sub.0] + [m.summation over (i=1)] [b.sub.i][x.sub.i] + [m.summation over (1=1)][m.summation over (j=1)] [b.sub.ij][x.sub.i][x.sub.j] + [m.summation over (i=1)] [m.summation over (j=1)] [m.summation over (k=1)] [b.sub.ijk][x.sub.i][x.sub.j][x.sub.k] (1)

where X([x.sub.1], [x.sub.2],.,xm) is the input variables vector, m is the number of input variables, B([b.sub.1], [b.sub.2], ..., [b.sub.m]) is the vector of coefficients. Each coefficient in Eq. (1) may be estimated to identify the system. The number of coefficients to be estimated, however, rapidly increases as the system increases in complexity and therefore the calculation will be difficult, with a large number of inputs and outputs data required. The GMDH is an identification technique which has been developed for such solution and which successively approximates the model to the complete description, using partial descriptions. Regression equations obtained by stepwise regression procedure are used as the partial descriptions, while in the basic GMDH; the following second order polynomial is used.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

where [y.sub.k] denotes the intermediate variable, [x.sub.i] and [x.sub.j] are input variables. Hence, this method builds a multilayered perception-type network for obtaining a polynomial description of stochastic system, Fig. 1. The GMDH algorithm proceeds as follow: Step (1); Select the input variables considered to affect the output. Then convert the raw input and output into normalized deviations from the average values. Step (2); Separate the data into a training set and a checking set. The training data are used to estimate the coefficients of the partial descriptions and the checking data are used to evaluate the accuracy of the partial descriptions and to prevent over fitting. Step (3); Form the partial descriptions using the stepwise regression procedure with all inputs taken two at a time. All combinations of r input variables are generated before learning each layer. The number of combinations is

[sup.m][C.sub.r] = m!/r!(m - r)! (3)

where m is the number of input variables and r is the number of inputs for each node (usually set to two according to the basic model introduced by Ivakhnenko [11]. Step (4); calculate the error criterion between each intermediate variable and checking data. By applying an error criterion at each layer, those variables which are least useful for predicting the correct output are filtered out. Step (3) and Step (4) are repeated until the lowest overall error criterion value (based on checking data set) at a certain layer is obtained. When one or more sets of new input and output data are given and it becomes necessary to renew the model to the new data, it can be possible to derive from the work performed by Uemasu and Mohri [7] an algorithm which avoids to forming a new model by means of GMDH, but from the original structure of the obtained model only its coefficients are adjusted with given appropriate weights to the newly given process data. This algorithm is called modified GMDH. This approach was proven to be computationally effective and memory economical and suitable for the situation where the system only changes gradually as in progressive tool wear. The scheme of building up a generation of high order variables is shown in Fig. 2.

2.1. Construction of training and checking set

The objectiveness of GMDH algorithm is based on the utilization of an external criterion to select the optimal model, which requires the data partition. The requirement of splitting data into two groups will lead to different models for different subsamples and researchers have investigated a number of techniques to overcome this situation as reported by Anastasakis and Mort [12]. The most used technique in the machining field is to separate the data points into training set and checking set according to the criterion of variance defined by the Eq. (4), where the variance of each data point, D(k), is calculated as

D(k)= [[[y.sub.j] - [[bar.y].sub.j]/[[bar.y].sub.j].sup.2]; j = 1,2,...,p; k =1,2, ..., N (4)

where p represents the range of data.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

The points with the larger variance are put into training set and the others into the checking set. The data into the training and checking set are separated using a table of random numbers.

2.2. Selection of effective variables

There are different criteria for screening out the least effective variables at each layer such as the regularity criterion, the unbiased criterion, the combined criterion and PRESS criterion [12]. These criteria usually take the common form containing two parts, a cost function which penalize the addition term in each layer, and another representing the mean squared error from regression. Any variable satisfying the criterion enters the next layer automatically. The criterion is evaluated using the checking set. This stage makes the number of retained variables at the output decrease from layer to layer thus the GMDH procedure doesn't become unstable as the process continues.

3. Tool flank wear modeling

3.1. Experimental setup

A series of experiments were carried out during machining of 80 mm diameter C20 steel bars on a machine tool installed on an appropriate elastic foundation designed to absorb vibrations produced by dynamical forces generated during the cutting process. The flank wear process of a triangular tungsten carbide WNMG tip with 5[degrees] rake angle is dealt with. Fig. 3 shows a part of the experimental setup. To measure the flank wear VB, the procedure implementation is based on using a Nikon profile projector. It is an optical device that allows accurate observation of the surface and the contour of opaque parts. The magnified image is projected onto the screen with perfect amplification. The measurement method is based on the use of cross-line screen. Fig. 4 illustrates the principle of the measurement technique of the tool flank wear VB. Details of the experiments for measurement of the tool flank wear have been given by Kara and al. in the reference [13].

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

3.2. Design of experiment

The cutting process in turning operation depends on many system parameters whose complex interactions make it difficult to describe the system mathematically. From literature of machining processes it can be deduced easily that the cutting regime variables (speed, feed and depth of cut), the chemical and mechanical properties of the tool, the geometric parameters of the tool, the lubrication quality, the system dynamics, etc.., can be considered candidates as input variables to the causality relationship. Tool flank wear and work-piece surface quality can be considered as the output parameters.

In this paper only the machining parameters were set during experimentation to examine the cutting regime influence on the tool flank wear. This data set constituted the input to the self-organizing network and consisted of three inputs and one output. The first independent input is the cutting speed (V). In this study five values in the range from 64 m/min to 237 m/min are considered. The second parameter is the feed (f). Five feeds are adopted in this study, ranging from 0.08 mm/rev to 0.2 mm/rev. The third parameter is the depth-ofcut (d) and five values are ranging between 0.5 mm to 1.5 mm. The cutting conditions used in this work are summarized in Table 1. Each parameter had five levels selected from practice. Evidently, the influence of system vibrations during cutting process on the obtained experimental results is minimized with special care taken experimentally to avoid this influence. Also, no lubrication was used.

3.3. Experimental results and discussions

A three-factor, five-level central composite rotatable design was used in this study. Table 2 shows the arrangement and the results of the twenty experiments carried out in this investigation. The range of each parameter was coded in five levels selected as follows

[-1.5 -1 0 + 1 + 1.5]

The targets for the tool wear are given in the last column of the Table 2. The tool flank wear data were carried out at the same cutting length for each trial cutting. The output of the GMDH reported in this paper is used to develop the mathematical model of the tool flank wear in next section. At least three tests at the each point of the design matrix were carried out.

4. Mathematical model for the tool flank wear

The model determination was carried out using the simplified algorithm of GMDH. A data sample is divided into two parts. The criterion MSE (Mean Square Error) is used, then approximately two-thirds of tests forms the training subset, and the remaining part of observations (e.g. every third point with the same variance) forms the checking subset. The training subset was used to derive estimates for the coefficients of the polynomial, and the checking subset was used to select the optimal model, that is one for which the regularity criterion MSE assumes its minimum

MSE = 1/[N.sub.b][NB.summation over (1)][[[y.sub.i] - [[??].sub.i]].sup.2] [right arrow] min (5)

where [N.sub.B] represents the range of checking subset data.

All input and output data in Table 2 are used in the form of the normalized deviations from average values by

[[xi].sub.i] (h) = [x.sub.i](h) - [[bar.x].sub.i]/[[bar.x].sub.i]; [PSI](h) = [y.sub.i](h) - [[bar.y].sub.i] [[bar.y].sub.i] (6)

where i (i=1,2,3) and h (h=1,2,3,4,5) denote the input variables and the experimental values number respectively. In this study, the algorithm allowed generating a model after three layers in the network. All possible pairs of input or output from previous layer are considered. Based on the outputs of the basic GMDH, the tool flank wear for the turning operation was modeled as

y = [b.sub.0] + [b.sub.1][x.sub.1] + [b.sub.2][x.sub.2] + [b.sub.3][x.sub.3] + [b.sub.4] [x.sup.2.sub.1] + [b.sub.5][x.sup.2.sub.2] + [b.sub.6][x.sup.2.sub.3] + +[b.sub.7][x.sub.1][x.sub.2] + [b.sub.8][x.sub.1][x.sub.3] + [b.sub.9][x.sub.2][x.sub.3] (7)

where [x.sub.1], [x.sub.2], [x.sub.3] are the normalized speed, feed and depthof-cut respectively. The GMDH network performed in this work found the ten coefficients {20.247, -0.920, -69.295, -20.513, 0.008, 46.273, 4.320, 1.070, 0.344, and 36.033} leading to the predictive model of the tool flank wear

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

The minimal weighted, training and testing-errors at different layers are shown in Table 3. They may be derived from the MSE criterion.

Fig. 5 shows the performance index on training and testing data for different layers. It could be observed that the error level drops along the network. Fig. 6 shows the GMDH prediction and the corresponding error. The error level clusters are mainly within the range 2.77% to 22.13%. As it could be observed, the prediction values closely follow the experimental results, except for trial number 9 where there is some noticeable deviation.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The application of the modified GMDH approach with adding a new set of data shown on the Table 4 from line 21 to line 28, provides better results as illustrated by the Fig. 7 where the error level is in a reduced range of 0.1% to 9.27% including trial n[degrees]9. The new predictive model with adjusted coefficients is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The effect of the speed, feed and depth-of-cut increase on the tool flank wear is clearly shown on the Figs. (8)-(10) where the distribution of tool flank wear according to the input parameters illustrates that when speed, feed and depth-of-cut augment, the tool wear increases. We note the strong influence of speed on the tool flank wear. Figs. (11)-(13) represent the first partial derivatives of the model described by the Eq. (9) according to the speed; feed and depth-of-cut respectively.

All these derivates increase linearly. Note them as

S = [partial derivative](VB)/[partial derivative]V; [S.sub.d] = [partial derivative](VB)/[partial derivative]d.

We observe that SV does not increase substantially in comparison with [S.sub.f] and [S.sub.d] in our experimental range conditions. This result reflects the low sensitivity of the wear acceleration according to the cutting speed level. Inversely the [S.sub.f] and [S.sub.d] are sensitive to the feed and depth-ofcut parameters due to the chip section variation which increases the abrasive wear phenomenon. So, we can conclude that the tool flank wear is strongly affected by the cutting speed parameter but with practically the same wear process acceleration at each level of the cutting speed. The explanation of this phenomenon can be given by the fact that flank wear is not strongly related to thermal phenomena generated by the increase in cutting speed than the me chanical abrasion caused by the chip section (defined by depth-of-cut and feed) on the tool flank even without lubrication.

5. Optimizing the tool wear model in turning operation

Several optimization techniques (genetic algorithm, particle swarm optimization, etc,...) could be used to further solve the problem of Eq. (9) subject to the machining constraint given as 46 [less than or equal to] [x.sub.1] [less than or equal to] 237; 0.08 [less than or equal to] [x.sub.2] [less than or equal to] 0.2; 0.5 [less than or equal to] [x.sub.3] [less than or equal to] 1.5.

This solution will give the optimal values for the response and input parameters. A number of optimization techniques are available that can easily solve Eq. (9) optimally. Here we apply the recent developed technique VNS (Variable Neighborhood Search).

5.1. VNS methodology

The basic idea of VNS metaheuristic is to use more than one neighborhood structure and to proceed to a systematic change of them within a local search.

The algorithm remains in the same solution until another solution better than the incumbent is found and then jumps there.

Neighborhoods are usually ranked in such a way that intensification of the search around the current solution is followed naturally by diversification. The level of intensification or diversification can be controlled by a few easy to set parameters. We may view the VNS as a 'shaking' process where a movement to a neighborhood further from the current solution corresponds to a harder shake.

Unlike random restart, the VNS allows a controlled increase in the level of the shake. Let us denote by Nk; k = 1,...,[K.sub.Max] a finite sequence of preselected neighborhood structures, and by Nk(x) the set of feasible solutions corresponding to neighborhood structure Nk at the point x, where x is an initial solution. Let us note that most local search metaheuristics use one neighborhood structure; i.e. [K.sub.Max] = 1. The following algorithm presents steps of the basic VNS heuristic.

Repeat until the stopping criterion is met:

(1) Set k[right arrow]1;

(2) Until k > [K.sub.Max] repeat the following steps:

(a) Shaking: generate a point x' at random from Nk(x);

(b) Local search: Apply some local search method with x' as the initial solution; denote by x'' the so obtained local minimum;

(c) Move or not: If x" is better than the encumber move there (x' [left arrow] x") and set k [left arrow] 1; otherwise k [left arrow] k + 1.

The stopping criterion may be e.g. the predetermined maximal allowed CPU time, the maximal number of iterations, or the maximal number of iterations between two improvements.

Let us note that the point x' is generated in step (2) at random in order to avoid cycling which might occur if any deterministic rule was used.

[FIGURE 8 OMITTED]

[FIGURE 9 OMITTED]

[FIGURE 10 OMITTED]

[FIGURE 11 OMITTED]

[FIGURE 12 OMITTED]

[FIGURE 13 OMITTED]

5.2. Results

For the tool wear, the optimal solution for Eq. (9) is given as shown in Table 5. The optimizer found optimal values of speed [x.sub.1] = 60 m/min; [x.sub.2] = 0.128 mm/rev; [x.sub.3] = 1 mm and VB = 9.01 urn These inputs values are the best ones to achieve the minimum wear possible in our turning range regimes. Using an optimization technique gives the best possible turning conditions and consequently such approach is extremely useful in a realizing a computer-aided process-planning system in a manufacturing environment.

5.3. Observations

Observing the experimental results from Tables 3 and 4, it is indicated that the trial No. 9 provides a value of VB = 9 closer to that found using the optimization tool of the model represented by Eq. (9). This experimentation shows that both GMDH and VNS methodology agree in their solutions. However, the GMDH method does not have a property that confirms that it has found a global minimum value. By using VNS methodology, we have confirmed that our GMDH approach can find a global minimum condition. Also, it is interesting to observe that the GMDH method uses only input values used for experimentation but the optimization technique such as VNS can find input values that were not used for the initial experiments. This is one difference between the solution realized using GMDH and other optimization techniques.

6. Conclusion

In this study, a GMDH algorithm is performed for modeling flank wear of a tungsten carbide tip as a function of the cutting speed, feed, and depth-of-cut during a turning operation. Firstly, the modeling methodology is presented, and then we perform a predictive model of the problem being solved in the form of a second-order polynomial based on the input variables. A modified GMDH algorithm is then applied to ensure robust results. The retained model seems with a great predictive capacity because the data obtained outside the experimental results have been well predicted. Also, the performed model indicates that cutting speed influences strongly the tool flank wear compared to the feed or the depth-of-cut. Moreover, this approach makes it easy to present the realized solution in a form that could be further optimized for the input parameters; i.e. the best cutting regime. Hence, the recent VNS approach is applied successfully in this work.

References

[1.] Astakhov, V.P.; Galitsky, V. 2005. Tool life testing in gundrilling: an application of the group method of data handling (GMDH), Int. J. of machine tools & Manufacture 45: 509-517. http://dx.doi.org/10.1016/j.ijmachtools.2004.09.003.

[2.] Ivakhnenko, A.G. 1968. The group method of data handling - A rival of the method of stochastic approximation, Soviet Automatic Control c/c of Avtomatika 13(3): 43-55.

[3.] Ivakhnenko, A.G.; Ivakhnenko, G.A. 1995. The review of problems solvable by algorithms of the group method of data handling (GMDH), Pattern Recognition and Image Analysis 5(4): 527-535.

[4.] Nagasaka, K.; Hashimoto, F. 1981. Estimation of quality of chip proposal by GMDH of variable selection type, Int. J. Mech. Tool Des. 21(%): 271-277.

[5.] Yoshida, T.; Nagasaka, K.; Kita, Y.; Hashimoto, F. 1986. Identification of a grinding wheel wear equation of the abrasive cut-off by the modified GMDH, Int. J. Mech. Tool Des. 26(3): 283-292. http://dx.doi.org/10.1016/0020-7357(86)90006-5.

[6.] Nagasaka, K.; Yoshida, T.; Kita, Y.; Hashimoto, F. 1987. Optimum combination of operating parameters in abrasive cut-off, Int. J. Mech. Tool Manufact. 27(2): 167-179. http://dx.doi.org/10.1016/S0890-6955(87)80048-2.

[7.] Uematsu, T.N.; Mohri, N. 1986. Prediction and detection of cutting tool failure by modified group of data handling, Int. J. Mach. Tool Des. Res. 26(1): 69-80. http://dx.doi.org/10.1016/0020-7357(86)90197-6.

[8.] Jiaa, C.L.; Dornfeld, D.A. 1998. A self-organizing approach to the prediction and detection of tool wear, ISA Transactions 37: 239-255. http://dx.doi.org/10.1016/S0019-0578(98)00035-4.

[9.] El-Khabeery, M.M.; El-Axir, M.H. 2001. Experimental techniques for studying the effects of milling roller-burnishing parameters on surface integrity, Int. J. of machine tools & Manufacture 41: 1705-1719. http://dx.doi.org/10.1016/S0890-6955(01)00036-0.

[10.] Onwubolu, G.C.; Buryan, P.; Lemke, F. 2007. Modeling tool wear in end-milling using enhanced GMDH learning networks, Int. J. Adv. Manuf. Technol. 39(11):1080-1092 http://dx.doi.org/10.1007/s00170-007-1296-1.

[11.] Ivakhnenko, A.G. 1971. Polynomial theory of complex systems, IEEE Trans. Syst. Man Cybern., SMC-1: 364-378.

[12.] Anastasakis, L.; Mort, N. 2001. The development of self-organization techniques in modeling: A review of the group method of data handling (GMDH), Department of Automatic Control &Systems Engineering, University of Sheffield, Research report N[degrees] 813. http://dx.doi.org/10.1109/TSMC.1971.4308320.

[13.] Kara-Ali, D.; Liazid, A.; Cheikh, A.; Hadjoui; F. 2007. Influence des parametres de coupe sur l'usure d'une plaquette en carbure metallique, COST: Revue Technologique et Scientifique 5: 39-46.

D. Kara Ali *, M.E.A. Ghernaout *, S. Galiz **, A. Liazid **

* Mechanical Engineering Department, Abou Bakr Belkaid University; B.O. 119; Route de Chetouane; 13000-Tlemcen-Algeri, E-mail: dj_kara_ali@yahoo.fr

** LTE-Laboratory; ENSET-Oran; B.O. 1523; El'Mnaouer 31000-Oran-Algeria, E-mail: ab_liazid@hotmail.com

http://dx.doi.org/10.5755/j01.mech.18.5.2696

Received April 01, 2011

Accepted October 12, 2012

Table 1
Summary of cutting conditions

Parameters Symbol Levels in code form

 -1.5 -1 0 +1 +1.5

Cutting speed, [x.sub.1] 64 89 125 173 237
 m/min
Feed, mm/rev [x.sub.2] 0.08 0.10 0.128 0.150 0.200
Depth, mm [x.sub.3] 0.5 0.75 1 1.25 1.50
Turning No lubrication
 operation
 condition

Table 2
Experimental results

 Speed V, Feed f,
 m/min mm/rev

 Actual Coded Actual Coded

1 89 -1 0.1 -1
2 173 +1 0.1 -1
3 89 -1 0.1 -1
4 173 +1 0.1 -1
5 89 -1 0.15 +1
6 173 +1 0.15 +1
7 89 -1 0.15 +1
8 173 +1 0.15 +1
9 64 -1.5 0.128 0
10 237 +1.5 0.128 0
11 125 0 0.128 0
12 125 0 0.128 0
13 125 0 0.08 -1.5
14 125 0 0.2 +1.5
15 125 0 0.128 0
16 125 0 0.128 0
17 125 0 0.128 0
18 125 0 0.128 0
19 125 0 0.128 0
20 125 0 0.128 0

 Depth d, Measured flank
 mm wear VB, [micro]m

 Actual Coded

1 0.75 -1 18
2 0.75 -1 126
3 1.25 +1 28
4 1.25 +1 152
5 0.75 -1 21
6 0.75 -1 135
7 1.25 +1 32
8 1.25 +1 160
9 1 0 9
10 1 0 309
11 0.5 -1.5 45
12 1.5 +1.5 80
13 0.5 0 40.5
14 0.5 0 50
15 0.5 0 47
16 0.5 0 45
17 0.5 0 46.5
18 0.5 0 46
19 0.5 0 48
20 0.5 0 45.5

Table 3
Minimal errors at different layers

 Minimal Minimal Minimal
 weighted training error testing error
 error

Layer1 0.0892427 0.1386601 1.889544
Layer2 0.0563340 0.05288310 0.4065458
Layer3 0.0712320 0.03398841 0.1564479

Table 4
Added set of the new input and output data

Trial Speed V, Feed f, Depth d,
no. m/min mm/rev mm

1 89 0.1 0.75
2 173 0.1 0.75
3 89 0.1 1.25
4 173 0.1 1.25
5 89 0.15 0.75
6 173 0.15 0.75
7 89 0.15 1.25
8 173 0.15 1.25
9 64 0.128 1
10 237 0.128 1
11 125 0.128 0.5
12 125 0.128 1.5
13 125 0.08 0.5
14 125 0.2 0.5
15 125 0.128 0.5
16 125 0.128 0.5
17 125 0.128 0.5
18 125 0.128 0.5
19 125 0.128 0.5
20 125 0.128 0.5
21 75 0.1 1
22 100 0.128 1.25
23 150 0.2 0.75
24 200 0.15 0.5
25 75 0.2 1.5
26 75 0.128 1.5
27 100 0.08 1
28 150 0.08 0.75

Trial Measured wear VB, Basic Modified
no. [micro]m GMDH GMDH

1 18 17.500 18.500
2 126 146.944 127.028
3 28 28.674 29.124
4 152 172.566 152.520
5 21 20.727 21.616
6 135 154.665 134.869
7 32 32.801 33.146
8 160 181.187 161.267
9 9 5.224 9.352
10 309 345.854 300.857
11 45 53.884 47.632
12 80 89.624 82.639
13 40.5 49.464 43.223
14 50 60.915 54.637
15 47 53.884 47.632
16 45 53.884 47.632
17 46.5 53.884 47.632
18 46 53.884 47.632
19 48 53.884 47.632
20 45.5 53.884 47.632
21 14 13.883
22 42 42.233
23 101 100.812
24 177 177.279
25 30 30.452
26 25 24.889
27 32 31.559
28 85 84.936

Table 5
Optimal solution using optimization technique

 Objective [x.sub.1] [x.sub.2] [x.sub.3]
 function VB

GMDH 9.35 64 0.128 1
VNS 9.01 60 0.128 1