【正文】 ould also be observed that the surface roughness was a minimum at the 250 m/min speed, 200 mm/min feed rate, 10? radial rake angle and mm nose radius. In order to understand the process better, the experimental results can be used to develop mathematical models using RSM. In this work, a mercially available mathematical software package (MATLAB) was used for the putation of the regression of constants and exponents. The roughness modelUsing experimental results, empirical equations have been obtained to estimate surface roughness with the significant parameters considered for the experimentation . cutting speed, feed rate, radial rake angle and nose radius. The first order model obtained from the above functional relationship using the RSM method is as follows:The transformed equation of surface roughness prediction is as follows:Equation 10 is derived from Eq. 9 by substituting the coded values of x1, x2, x3 and x4 in terms of ln s, ln f , lnα and ln r. The analysis of the variance (ANOVA) and the Fratio test have been performed to justify the accuracy of the fit for the mathematical model. Since the calculated values of the Fratio are less than the standard values of the Fratio for surface roughness as shown in Table 4, the model is adequate at 99% confidence level to represent the relationship between the machining response and the considered machining parameters of the end milling process.The multiple regression coefficient of the first order model was found to be . This shows that the first order model can explain the variation in surface roughness to the extent of %. As the first order model has low predictability, the second order model has been developed to see whether it can represent better or not.The second order surface roughness model thus developed is as given below:where Y2 is the estimated response of the surface roughness on a logarithmic scale, x1, x2, x3 and x4 are the logarithmic transformation of speed, feed, radial rake angle and nose radius. The data of analysis of variance for the second order surface roughness model is shown in Table 5.Since F cal is greater than , there is a definite relationship between the response variable and independent variable at 99% confidence level. The multiple regression coefficient of the second order model was found to be . On the basis of the multiple regression coefficient (R2), it can be concluded that the second order model was adequate to represent this process. Hence the second order model was considered as an objective function for optimization using genetic algorithms. This second order model was also validated using the chi square test. The calculated chi square value of the model was and them tabulated value at χ2 is , as shown in Table 6, which indicates that % of the variability in surface roughness was explained by this model.Using the second order model, the surface roughness of the ponents produced by end milling can be estimated with reasonable accuracy. This model would be optimized using genetic algorithms (GA). The optimization of end millingOptimization of machining parameters not only increases the utility for machining economics, but also the product quality toa great extent. In this context an effort has been made to estimate the optimum tool geometry and machining conditions to produce the best possible surface quality within the constraints.The constrained optimization problem is stated as follows: Minimize Ra using the model given here:where xil and xiu are the upper and lower bounds of process variables xi and x1, x2, x3, x4 are logarithmic transformation of cutting speed, feed, radial rake angle and nose radius.The GA code was developed using MATLAB. This approach makes a binary coding system to represent the variables cutting speed (S), feed rate ( f ), radial rake angle (α) and nose radius (r), . each of these variables is represented by a ten bit binary equivalent, limiting the total string length to 40. It is known as a chromosome. The variables are represented as genes (substrings) in the chromosome. The randomly generated 20 such chromosomes (population size is 20), fulfilling the constraints on the variables, are taken in each generation. The first generation is called the initial population. Once the coding of the variables has been done, then the actual decoded values for the variables are estimated using the following formula:where xi is the actual decoded value of the cutting speed, feed rate, radial rake angle and nose radius, x(L) i is the lower limit and x(U) i is the upper limit and li is the substring length, which is equal to ten in this case.Using the present generation of 20 chromosomes, fitness values are calculated by the following transformation:where f(x) is the fitness function and Ra is the objective function.Out of these 20 fitness values, four are chosen using the roulettewheel selection scheme. The chromosomes corresponding to these four fitness values are taken as parents. Then the crossover and mutation reproduction methods are applied to generate 20 new chromosomes for the next generation. This processof generating the new population from the old population is called one generation. Many such generations are run till the maximum number of generations is met or the average of four selected fitness values in each generation bees steady. This ensures that the optimization of all the variables (cutting speed, feed rate, radial rake angle and nose radius) is carried out simultaneously. The final statistics are displayed at the end of all iterations. In order to optimize the present problem using GA, the following parameters have been selected to obtain the best possible solution with the least putational effort:Table 7 shows some of the minimum values of the surface roughness predicted by the GA program with respect to input machining ranges, and Table 8 shows the optimum machining conditions for the corresponding minimum values