Genetic algorithms solution to the single- objective machining process optimization time model

Minimum Production Time model of the machining process optimization problem comprising seven lathe machining operations were developed using Genetic Algorithms solution method. The various cost and time components involved in the minimum production cost and minimum production time criteria respectively, as well as all relevant technological/practical constraints were determined. An interactive, user-friendly computer package was then developed in Microsoft Visual Basic.Net environment to implement the developed models. The package was used to determine optimal machining parameters of cutting speed, feed rate and depth of cut for the seven machining operations with twenty-three technological constraints in the conversion of a cylindrical metal bar stock into a finished machined profile. The result of the single-objective machining process optimization models shows that the minimum production time is 21.84 min.


INTRODUCTION
Taylor first realized the importance of machining optimization [1] in his pioneering work "On the Art of Cutting Metals". Since then, optimization of machining processes remains an ongoing activity, as evidenced by the optimization studies that were carried out over the last century [2]. In machining process optimization criteria are usually based on three objectives of: the minimum total cost per component; the maximum production rate; and the maximum profit-rate criterion [3][4][5].
Selection of cutting parameters is usually a difficult task, where the following aspects are required: knowledge of machining; empirical equations relating the tool life, forces, power, surface finish, etc., to develop realistic constraints; specification of machine tool capabilities; development of an effective optimization criterion; and knowledge of mathematical and numerical optimization techniques [6,7].
Several optimization techniques have been employed for machining process optimization since the introduction of computers to machining systems.
Linear programming was used for machining process optimization [8][9][10] developed a Nelder-Mead simplex method to determine the optimum machining conditions. In most of the works above, the problems were simplified by considering only one or two variables such as the cutting speed and feed rate, in order to optimize the economical machining performance. They assumed that a single cut can achieve the required maximum metal removal rate (MRR).
Geometric Programming (GP), one of the nonlinear optimization techniques, has been extensively adopted [11], in which the constrained models are converted into a dual geometric programming formulation and then into an unconstrained nonlinear programming formulation.
Traditional non-linear optimization techniques have also been extensively used. Wen et al [12] adopted the successive quadratic programming method to solve the non-linear off-line optimization scheme for a surface grinding process. Xiao et al [13] applied an iterative Newton's method for a non-linear internal cylindrical plunge grinding process. Jha and Hornik [14] used the generalized reduced gradient method to optimize the tool geometry and cutting conditions in plain milling process.
Sonmez and Baykasoglu [6] outlined the development of an optimization strategy to determine the optimum cutting parameters for multi-pass milling operations such as plain milling. The developed strategy was based on the maximum production rate and incorporated eight technological constraints. The optimum number of passes was determined via dynamic programming and the optimum values of cutting conditions were found based on the objective function by using the geometric programming technique. Jang [15] developed a unified optimization approach for the selection of the machining parameters (cutting speed, feed, and depth of cut) to provide the maximum metal removal rate. Ermer [16] analyzed a nonlinear objective function with inequality constraints to determine the optimal machining conditions by geometric programming. Lambert and Walvekar [17] developed a dynamic programming model for the multi-pass turning operation under constraints of force, cutting power and surface finish to determine values of machining variables and minimum production cost. They considered two-pass turning operations. Shin and Joo [18] presented a model for the multi-pass turning operation using a fixed machining interval. They used dynamic programming for the selection of depth of cut for individual passes. The final finish pass was fixed based on the minimum allowable depth of cut and the remaining depth of cut was divided into a number of rough passes of equal sizes to obtain the minimum total cost. Lee et al [19] developed a fuzzy non-linear programming model to optimize machining operations. The model was used to select the toolholder, insert and cutting conditions (feed, speed and depth of cut). They used dynamic programming to select optimal cutting conditions. The traditional non-linear optimization techniques are mostly gradient-based and possess many limitations in application to today's complex machining models. Secondly, they cannot deal with integer/discrete design variables directly; integer design variables have to be approximated from continuous values. Therefore, one must resort to nonsystematic optimization techniques, such as Evolutionary Algorithm.
Groover [20] used Monte Carlo simulation to study the machining economic problem considering tool wear and surface roughness. Dereli et al [21] explained the application of Genetic Algorithms (GAs) for determination of optimal sequence of machining operations based on either minimum tool change or minimum tool traveling distance or safety. Srikanth and Kamala [22] applied a Real Coded Genetic Algorithm (RCGA) to determine minimum surface roughness values, and their corresponding optimum cutting parameters, for turning process. But they only considered four constraints. Saravanan et al [23] showed an optimization method for cutting conditions in continuous profile machining in order to minimize the production cost. For the optimization method, they used Genetic Algorithms (GAs) and Simulated Annealing (SA) and compared the results. Amiolemhen and Ibhadode [24] applied Genetic Algorithms (GAs) to determine the optimal machining parameters in the conversion of a cylindrical bar stock into a continuous finished profile using the minimum production cost criterion. They developed single and multi-pass models for seven machining processes involved in continuous prolife machining.
Genetic Algorithms solution which has been used extensively in non-linear machining optimization problems [25][26][27][28][29], is the choice for this research work.
Hence, this paper employs Genetic algorithms to determine the minimum production time in the conversion of cylindrical bar stock into a continuous finished part. A user friendly and iterative computer package developed in the Microsoft Visual Basic.Net environment is employed to determine the optimal machining parameters for machining a continuous finished profile from bar stock.

Machining process optimization models
Mathematical models have been developed for the following machining processes involved in the conversion of a cylindrical bar stock into a continuous finished part. These machining processes are: facing; turning; centreing; drilling; boring; chamfering; and parting. The models for the above machining operations are summarized in Table 1. The time model for each cutting operation is minimized subject to the constraints specified by the given equations.
These equations are given below:  bounds on cuttings peed: finishing:  bounds on feed rate: finishing:  bounds on depth of cut: roughing: finishing:  tool Life constraint: roughing: finishing:  cutting force constraint: roughing: finishing:  cutting power constraint: roughing: finishing:  chip-tool interface temperature constraint: roughing: finishing:  dimensional accuracy constraint: roughing: finishing:  stable cutting region constraint: roughing finishing:  surface finish constraint: finishing:  miscellaneous constraints: Finishing cutting speed: Finishing Feed rate: Finishing depth of cut: Total depth of cut constraint: d s = d t -n,  bounds on number of rough cuts:   3, 5, 7, 9, 11, 13 7 Chamfering

Optimization by Genetic Algorithms
Outline of the Basic Genetic Algorithm

The Genetic Algorithms procedure
Generate initial population a. Determine population size The population size used in this work is, np = 20, in accordance with the recommendation of Schaffer [30].

b. Initialisation
The solution space of the population size, np = 20 is generated randomly between the bounds of each decision variable. In this work the decision variables are cutting speed, feed rate and depth of cut.

I.
Choosing solution representation The string of bits or genes in the chromosome could be binary, real integer number, etc [31]. In this work, binary string format of finite length was adopted.
II. Determination of chromosome lengths The total length of each design variable represented in a binary string is determined as follows:  Choosing level of precision The level of precision or the number of decimal places of each decision variable, p = 4 was adopted. Since the objective function is a minimization problem, the fitness function of the i th solution is thus evaluated by: where: g max (x) is the maximum objective function value and g i (x) is the objective function value of the i th solution.

Creation of a new population
After the transformation of the integer values into binary strings, Genetic Algorithms operators are applied. Here the three operators (reproduction, crossover, and mutation) are used.

a. Reproduction
The two chromosomes (strings) with best fitness and the second best fitness are allowed to live and produce offspring in the next generation, after evaluation. These chromosomes are the "elites chromosomes".

b. Selection and crossover
The cumulative probability is used to decide which chromosomes will be selected to crossover. The cumulative probability is calculated in the following steps:

I.
Selection of pairs of chromosomes for mating The Roulette wheel selection process was used selection and the cumulative probability, Cf i , is used to decide which parents will be selected for mating. And, the Roulette wheel is constructed as follows: Calculate fractional fitness (selection probability), Pf i , for each chromosome: Calculate the cumulative fitness (probability), Cf i , for each chromosome:

II.
Application of crossover operator to the selected pairs of chromosomes The crossover probability used is, P c = 0.80. Then, a random number was generated such that, r  (0, 1); and if r < P c , then crossover is carried out otherwise it is left unchanged.
c. Application of mutation operator to the reproduced chromosomes Mutation alters one or more genes with a probability equal to the mutation rate (of the order of 0.005 to 0.01). A random number is generated such that, r r < Pm, then that bit is complemented otherwise it is left unchanged.

d. Formation of a new population
After the mutation exercise, new strings are created which are then added to the two elite chromosomes from the initial population to form a new population.

a. Decoding the newly formed population
The newly formed chromosomes after the mutation operation are usually decoded as follows:

Evaluation of objective function values
The objective function values of the model being applied are determined using the newly formed population and then the results are checked for optimality.

Termination method
A new population is created as a result of completing one-iteration of the Genetic Algorithms. The iteration is terminated if optimum results are obtained; otherwise it is repeated until the maximum number of GA generation is reached [31].
In this work, the Genetic Algorithms procedure was terminated after 50 generations.

Implementation
The elements of the proposed models developed using Genetic Algorithm have been implemented in the software developed in Microsoft Visual Basic.Net environment and run on a Pentium 4 PC with 3.0 GHz Intel Processor and 2 GB of RAM. The values set for different parameters of the genetic algorithm are shown in Table 2

Illustrative example
An illustrative example has been adopted from [24], [29] to demonstrate the performance of the proposed models. Table 3 shows the data of the illustrative example.  Fig.1 contains the optimum results of the seven machining processes considered using the minimum production time model for the 50 population generations. This table also shows the optimum cutting parameters of the seven machining processes considered and the overall production time per workpiece. Fig. 2 shows the plots of maximum selection probability (fractional fitness) and corresponding minimum costs with respect to the number of generations. The fitness plot shows that the selection probability varies within the range of 0.063 -0.119. The time plot shows that time of turning is about 9.9 min/piece from the 1st to the 5th generations. It then drops to about 8.7 min/piece from the 6th generation and remains constant at this value to the 47th generation. Thereafter the cost drops to about 8.32 min/piece from the 48th to the 50th generations. These plots show that in the neighbourhood of a drop in fractional fitness as the number of generations increase, there is a corresponding drop in the turning time. Between the 5th and 6th generations the fractional fitness drops from 0.095 to 0.071 when the time correspondingly drops from 9.9 min/piece to 8.32 min/piece. Similarly between the 47th and 48th generations, the fractional fitness drops from 0.095 to 0.070 while the production time correspondingly drops from 8.68 min/piece to 8.32 min/piece. These points of reduction in cost with respect to corresponding drops in fractional fitness relate to when the Genetic Algorithms solution is being reset by the crossover and mutation operators. Fig. 3 shows the plot of minimum production time for the seven machining operations carried out on the workpiece as given by Table 11. The Figure shows that the production time drops rapidly from 38.02 min/piece from the 1st generation to 34.3 min/piece at the 2nd generation and 34.1 min/piece from the 3rd generation to 24.7 min/piece at the 4th generation, and then to 22.8 min/piece at the 7th generation. The time drops between the 1st and the 7th generations represent a time slope of 1.586 min/generation. Whereas that between the 7th generation (22.8 min/piece) to the 48th generation (28.1 min/piece) is 0.024 min/generation representing 66 times the time slope between the 1st and 7th generations. This goes to show how effective the GAs solution technique is, in quickly converging to the optimum value. Table 4 shows the optimum cutting parameters and optimum machining time of the seven machining processes considered using the minimum production time model as well as the overall production time per workpiece.

RESULTS AND DISCUSSION
Using the data supplied by Ibhadode [33], the developed models gave production time for the monoplex die shown in Fig. 5 as 415.13 min. The details of these results are shown in Table 5.
But, when the original Taylor's tool life was replaced with the modified Taylor's tool life, the production time became 360.25 min. These results are also shown in Table 5. Examination of the optimum solutions given in Table 5 has shown that for the two cases, the optimum production times for the monoplex die container are superior to the conventional recommended solutions (Ibhadode; 2009). While this was anticipated from the optimization analysis, it further confirms that the optimization models are reliable tools for application on the shop floor.

CONCLUSIONS
Single-objective machining process optimization models were developed for seven machining processes involved in the production of a monoplex die container using the minimum production time model and subject to 22 technological constraints. The proposed model when implemented in Genetic Algorithms methodology gave an optimum production time of 21.84min/workpiece.
The results show that the minimum production time models predict that turning, drilling and boring have the first, second and third highest production time components respectively, for the workpiece considered. Thus, the models suggest that turning, drilling and boring operations are very important operations which demand the most production resources for the workpiece under consideration. It is therefore very important to ensure that the optimum cutting parameters of cutting speed, feed rate and depth of cut are used as derived.
The models also show that the operations of centering, parting and chamfering require the least production resources for the workpiece considered.
A comparison of the models developed (in which the optimum cutting parameters are determined by applying GAs to the models) with the conventional method of using static cutting parameters showed that the models predict better production times. Thus, the models perform better than the conventional method.
A robust Genetic Algorithms solution that is fast and efficient was developed and used to implement the optimization models.

Acknowledgements
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