Stepwise approach for the evolution of generalized genetic programming model in prediction of surface finish of the turning process
Introduction
Components produced during the turning operation have critical features that require specific surface finish [1]. Surface roughness is a widely used index for the measure of product quality. Hence, achieving desired surface roughness is of prime importance for the functional behaviour of the component. Past studies reveal that surface roughness depends on process parameters such as tool geometry, cutting conditions and work piece properties. These process parameters may be optimised for obtaining minimum cost and minimum production time. However, for obtaining the optimal input process parameter settings, the surface roughness needs to be predicted accurately. Hence, the modelling of turning process has attracted a great community of researchers with the purpose of reduction of overall cost of the engineering component [2]. With the use of numerically controlled CNC machines, the need for process modelling and optimisation is strengthened.
Considerable amount of research has been done in the prediction of surface roughness of the turning process [3], [4], [5], [6], [7], [8], [9], [10]. Researchers have developed physics-based models to understand the behaviour of turning, but this may be a challenging task with the availability of partial information about the process [2], [11]. The empirical modelling based only on the given data is a possible route for the modelling of the process. For this purpose, several empirical modelling methods such as regression analysis, artificial neural networks (ANN), support vector regression (SVR), fuzzy logic (FL) and genetic programming (GP) have been extensively applied in the prediction of surface roughness [12], [13], [14], [15], [16].
The regression analysis is based on statistical assumptions, and thereby induces uncertainty in the prediction ability of the models [17], [18], [19]. These models cannot be used to generalise the process data. ANN is well known for capturing the dynamics of the process. However, the optimal architecture of ANN is determined either through trial-and-error approach or by hybridising it with heuristic optimisation methods such as genetic algorithm (GA), and particle swarm optimization. This indicates that some skills/knowledge is needed to select the optimal architecture of ANN for faster training and better accuracy of the model [20], [21], [22], [23], [24]. SVR based on principle of SRM, is known for injecting generalisation ability in the models. Least square-support vector machines (LS-SVM) variant of SVR has been used for predicting the performance of turning process [7], [25], [26]. However, it does not provide explicit formulation between the input and output process parameters, and gives output values in crisp form.
FL models mostly used for modelling turning process are Mamdani and adaptive neuro-fuzzy inference system [27], [28], [29], [30]. The formulation of FL model requires expert knowledge to formulate the fuzzy rules. Researchers have carried out additional set of experiments to test the empirical models, but it involves high labour costs and results in increase in overall cost of the product [31], [32].
The applications for the explicit formulation of the performance of machining process using evolutionary approach GP have been on the rise [33], [34], [35], [36], [37], [38]. The main advantage of GP over the regression analysis and other statistical modelling techniques is that it has the ability to generate mathematical equations without assuming any prior form of the existing relationships. GP and its variants have been successfully applied for modelling the performance of various non-conventional and machining processes [39], [40]. The performance attributes measured were cutting force, surface roughness, tool wear, etc. Other variant of GP that uses set of genes for the formulation of model is multi-gene genetic programming (MGGP).
The peculiarity of MGGP method [41], [42] is that each model participating in its evolutionary stage is the combination of several genes combined using the least squares method. The applications of MGGP method suggest that it performs better than the traditional GP method [43], [44], [45]. In traditional GP method, the model is a single tree/gene expression whereas in MGGP, the model formed is a linear combination of several trees/genes. Gandomi and Alavi [46], [47], [48] in his work demonstrated the usefulness of MGGP approach in designing the non-linear models based on the data obtained from the complex non-linear systems. Past studies reveal that MGGP method provides a fast and cost-effective explicit formulation of a mathematical model based on multiple variables with no existing analytical models. Despite remarkable capabilities of MGGP method [49], [50], [51], [52], [53], [54], it is found that during the combination mechanism using least squares method in its evolutionary stage, the occurrence of genes of lower performance can degrade the performance of the MGGP model [55], [56], [57], [58]. This limitation of the MGGP method has motivated us to develop modified MGGP (M-MGGP) method by sensibly selecting the relevant genes of higher performance for the combination mechanism.
In the present work, a M-MGGP method is proposed and applied to the modelling of surface roughness of the turning process. In this method, a stepwise regression approach is introduced for the combination of genes. Unlike least squares method, the stepwise approach selectively eliminates redundant/poor performing genes and thus only combines the high performing ones. One objective of the present work is to compare the performance of the M-MGGP model to those of the standardized MGGP approach, SVR and ANN. Sensitivity and parametric analysis is then conducted for the proposed model to accentuate the principle behind the process and examine the dominant input parameters.
Section snippets
Experimental details of turning process
In the present work, the turning phenomenon to be modelled is referred from an earlier study conducted on modelling and optimisation of hard turning of AISI H11 steel using response surface methodology [59]. The experiments were performed in dry conditions using lathe type SN40C with a spindle power of 6.6 KW. The sample material used was AISI H11 hot work steel which is often used for the manufacture of diecasting moulds, dies and helicopter rotor blades. The composition of AISI H11 steel is
Multi-gene genetic programming
Genetic programming (GP) is an advanced modelling method that generates models in the form of tree structures based on the principle of Darwinian natural selection [63]. The initial population of individuals is randomly created to search for the solutions in large global space. Each member of the population is a tree structure comprising functions and terminals. The functions and terminals are chosen from the function and terminal set respectively. A function set F usually consists of basic
Evaluation of models performance
The results obtained from the three modelling methods, MGGP, M-MGGP and ANN, are illustrated in Fig. 9, Fig. 10, Fig. 11, Fig. 12 on the training and testing data respectively. The best prediction method that gives good generalization ability is determined by comparing these modelling methods using the six metrics: the square of the correlation coefficient (R2), the mean absolute percentage error (MAPE), the RMSE, relative error (%) and multi-objective error functions (MO1 and MO2) [72] given
Sensitivity and parametric analysis of the proposed model
For the validation of the robustness of the M-MGGP model, sensitivity and parametric analysis about the mean is conducted. The sensitivity analysis (SA) percentage of the surface roughness to each input parameter is determined using the following formulas [73]:where and are, respectively, the maximum and minimum of the predicted output over the ith input domain, where other variables are equal to their mean values.
Table 9 shows the
Conclusions
We demonstrated the use of experimental and soft computing methods for studying the mechanism of turning of AISI H11 Steel. The present work addresses issues in procedure of formulation of the MGGP model. To counter this, M-MGGP methodology is proposed. The performance of the M-MGGP model is found to be better than those of the other three models. In addition, the M-MGGP method evolves models of lower complexity when compared to the models evolved from MGGP method. This concludes that we have
Acknowledgement
This work was partially supported by the Singapore Ministry of Education Academic Research Fund through research Grant RG30/10, which the authors gratefully acknowledge.
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