Multi Pass Optimization of Cutting Conditions by Using the Genetic Algorithms

Production of high-quality products with lower cost and shorter time is an important challenge to face of increasing global competition. Determination of optimal cutting parameters is one of the most important elements in any planning process of metal parts. In this study we present a multi-optimization technique based on genetic algorithms and dynamic programming, to search for optimal cuttings parameters such as cutting depth, feed rate and cutting speed of multi-pass turning processes. Two conflicting objectives, the production cost and operation time are simultaneously optimize under a set of practical of machining constraints. The proposed model deals with multi-pass turning processes in which the cutting operations are divided into multi-pass rough machining and finish machining. Results obtained from Genetic algorithms method are used to define the optimum number of machining passes by dynamic programming; such technique helps us in the decision making process. An example is presented to develop the procedure of this technique.


INTRODUCTION
In previous works (Assas andDjenane, 2001, 2003a), we have proposed an optimization method based on a combined criterion, by solving an optimization problem with a Bi-criteria objective function for a unique pass manufacturing in the case of turning.In this study, the previous approach is applied but for the optimization of conditions in multiple passes manufacturing.
In turning operations, a cutting process can possibly be completed with a single pass or by multiple passes.Multi-pass turning is preferable over single pass turning in the industry for economic reasons (Wang, 2007).The optimization problem of machining parameters in multi-pass turnings becomes very complicated when plenty of practical constraints have to be considered (Shutong and Yinbiao, 2011).Abuelnaga and El-Dardiry (1984) discussed a number of traditional optimization methods and highlighted the relative advantages and disadvantages of the methods for solving the problems of machining economics.These objectives can be represented in terms of the machining parameters such as cutting speed, feed, depth of cut and the number of passes.
During selection of the parameters, care must be taken to ensure that the essential constraints are satisfied.Cutting force, power, surface finish and tool life are some of the commonly considered constraints (Ermer, 1971;Shin and Joo, 1992;Al-Ahmari, 2001;Sankar et al., 2007).Initial research in machining process optimization focused on single pass operations (Taylor, 1906;Gilbert, 1950;Petropoulos, 1973;Abuelnaga and El-Dardiry, 1984).The mathematical programming techniques like linear programming (Ermer and Patel, 1974), graphical methods (Kiliç et al., 1993) and geometric programming (Ermer, 1971;Petropoulos, 1973) had been used to solve optimization problems of machining parameters in multi-pass turnings.But, these methods of optimization do not fare well over a broad spectrum of problem domains.Jawahir and Wang (2007) have summarized the recent contributions to the field of machining process modeling and optimization.They have also presented a machining process optimization method in which many process performances such as surface roughness, cutting force, tool life and material removal rate have been combined into a single objective using weight factors.
In this study, we use the method developed by Agapiou (1992a), based on the dynamic programming to which we have introduced modifications and developed a program for that purpose.The optimal value of passes, the speed, the advance and the cutting depth for each pass represent the decision variables.It is a four -variable problem in which the pass number and the cutting depth for each pass are determined by the dynamic programming procedure.The speed of optimal cutting as well the optimal advance for each pass is determined by using the combined criterion method.
Cutting process model: The concept of dynamic programming is very useful for the treatment of multipass problem.
The total cutting depth is divided into N sections equal to d: The decision variable which is the cutting depth, d c , to be removed by pass i is represented by d (i, j) and the variable of state which is the part diameter of the pass i is represented by D i .The d (i, j) means that the cutting depth begins at diameter D i and includes j sections of dimensions d.
Consequently, the total cutting depth D C is divided into N equal sections which are the N discrete decision states for the dynamic programming.

OPTIMIZATION OF THE NUMBER OF PASSES BY DYNAMIC PROGRAMMING
In general when the pass begins from the section i and ends at the section q the objective function is expressed by: With j = i -q, j is the number of size sections d.
The cost of production by pass is expressed by: The duration of production is expressed by: The cutting length is always constant for all manufacturing passes: Fig. 1: Schema for the decomposition of the total cutting depth in 8 sections for the dynamic programming The diameter for each pass is expressed by: For the multi-passes manufacturing where M passes are necessary to achieve the turning operation the total production cost and total production duration are expressed by: k is the index which indicates the number of any pass.The timing machine for each pass is: The steps of dynamic iterations are represented by an inferior triangular matrix with N×N dimension.The elements (i, j) of the matrix respectively represent the diameter of starting section and the number of manufactured sections for any kind of pass.
The schematic representation of the dynamic programming process is expressed by the diagram of Fig. 1, where D c is divided into eight equal sections: It is known that below a certain depth cutting value DMOP, for a specific couple tool-matter, the simple pass manufacturing is optimal.
The maximum value of the cutting depth DMAXP is also known.
The inferior triangular matrix for the example in Fig. 1, is expressed as following by: where X represents the values of objective function, the diagonal elements represent the possible simple passes.The elements: 21, 31 and 32 are nil because the cutting depth for one given pass d c is inferior to DMOP and the elements: 55, 65, 75, 77, 86, 87 and 88 are nil because the cutting depth for one donated pass dc is superior to DMAXP.
In the same procedure, the cutting speeds and the advances are also arranged in two inferior separated triangular matrixes We introduce the following notations: Using the strategy of dynamic programming, supposing that the optimum is obtained from the i ème section, (1<i<N) until the final diameter (Intern) of the 1 st section, the steps of dynamic programming are the followings: Step 1: For a cutting depth inferior to DMOP such as: d(i, j) = i.d≤DMOP, we calculate the objectives ''functions'' corresponding UOP(i) to one pass finishing operations and we continue with the following steps.
Step 2: For the depths of pass DMOP<i.d≤DMAX:we calculate the 'objectives' functions corresponding u(i, j) to one finishing pass operation.
Step 3: Next we evaluate the calculations of objective 'functions' for multi-pass operations such as: With: UOP (i-r) is the minimum objective function starting from the section (i-r) until the first section, u(i, r) is the objective function of one pass from the section (i) to the section (i -r) with a cutting depth d c = r.d.
The approach based on the dynamic programming is described by the algorithm of Fig. 2.
The mathematic formulation of these functions is as following: The optimum in section 1 to 3 is obtained with one pass, since it is supposed that: d(i, i) ≤ DMOP, the sections i = 1, 2, 3 form a single branch.
In the 4th section for i = 1, 2, 3 the results are formulated in four branches: The optimum is obtained by the equation: u(4, 3) represent the branch of the 4th section connected to the 1st section and UOP(1) corresponds to the branch of the 1st section which is connected to final diameter D 0 .
The same, the 7 th and the 8th sections are not connected to the 1st section and to D 0 Because d( 7 15) we obtain: The Eq. ( 16) gives the number of optimum passes M = 3, two passes for the draft operations and one pass for the finishing with the cutting depths respectively: d(8, 4) = 4d, d(4,3) = 3d, d(1, 1) = d, with the total cutting depth:

OPTIMIZATION BY GENETIC ALGORITHM
The GA are exploration algorithms based on the mechanisms of natural selection and genetics.They use at a time the survival principles of the best adopted structures.In each generation, a new set of artificial creatures (chains of characters) is created using parts of the best elements of the previous generation.Although using the hazard, the GA is not purely random.They effectively exploit information obtained previously to speculate on the position of new points to be explored, hoping to improve the performance (Assas and Djenane, 2003b;Djari et al., 2007;Sardiñas et al., 2006;Rao and Pawar, 2010).

General function of Ga:
Stages of GA (Assas and Djenane, 2003b;Djari et al., 2007).The steps of an AG are as follow (Fig. 3): • Creation of the initial population.
• Evaluation of each chromosome of the initial population.
• Selection and regrouping of the chromosomes per pairs.• Application of crossover and the mutation operators.• Evaluation of new chromosomes and insertion in the following population.• If the stopping criterion is reached, the genetic algorithm stops; otherwise the algorithm returns back at the stage 3.
From generation to generation, the size of population remains constant.
At a generation, the terminology used is that of genetics.
• For each variable xi we associate a gene • The chromosome is a set of genes; the chromosomes are the elements from which the solutions are elaborated • The reproduction is the step of chromosomes combination.The genetic mutation and the crossing are reproduction methods.

ELITISM SRATEGY
The strategy called Elitism consists in recopying the best chromosome of the current population on the population of the next generation.This latter is completed by other chromosomes, generated with a traditional manner until getting the needed number of individuals.So, it becomes impossible for the e best chromosome of the next generation to be inferior to that of the former generations.
The performances of genetic algorithm are largely ameliorated.

GA operators:
Crossing operator: This operator combines the chromosomes of two individuals to get two new.A crossing in one point consists of exchanging a fragment of two chromosomes.The couple being created, the process happens in two steps: Random choice of an identical cut-off point on both chromosomes.
Cutting of the two chromosomes (Fig. 4) and exchanging the two fragments located on the right.
The crossing"1 point" is the simplest and most classical for coding using a binary encoding.
An immediate generalization of this operator consists of multiplying the cut-off points on each genotype.The crossing "1 point" and "2 points" (Fig. 5) are frequently practiced for their simplicity and good efficiency.

Mutation operator:
We define a mutation as being the inversion of a bit in a chromosome (Fig. 6).Classically, the mutation operator modifies randomly the symbols of genotype with a weak probability, equal to the rate of mutation.On the other hand, the mutation ensures a random local research around each individual.
Following this order of idea, the mutation can considerably improve the quality of solutions discovered.

APPLICATION AND RESULTS OF OPTIMIZATION
To find the number of optimal passes, we have developed a program in Fortran language (Fig. 2).
In order to test the influence of sections number on the optimal value of pass number, we have divided the total cutting depth Dc = 10.16 mm successively into 4, 8, 10 and12 sections equals for the piece length L = 203 mm and one diameter D = 152 mm.The parameters of machining are mentioned in Table 1.
Table 2 represents the values of advance speed and cutting speed corresponding to different depths of cutting calculated by a genetic algorithm for draft operations SR max = 8 µm and finishing SF max = 2 µm.
The limitations on the constraints (Agapiou, 1992b) are the followings: • Limitations on the advance speed:  • Limitations on the cutting speed: V≥V min , V≤V max • Limitations on the cutting depth: dc≥dc min , dc≤dc max • Limitations on the maximum power allowed by the machine: 0.0373 V 0.91 f 0.78 d 0.75 ≤HP max • Limitations on the cutting effort: V -0.1013 f 0.725 d 0.75 ≤F max • Limitations on the surface state: 14.75V -1.52 f 1.004 d 0.25 ≤SR max • Limitations on the temperature of cutting: 74.96V 0.4 f 0.2 d 0.105 -17.8≤θ max The conditions of optimal manufacturing for the different numbers of sections 4, 8, 10, 12 are mentioned in Table 3.
The influence of the sections number N used by the technique of dynamic programming is observed by comparing the results of Table 3 for N = 4, 8, 10, 12.
By increasing the number of sections N from 4 to 8the timing cost of total production decreases from 12.16% and 5.54%, similarly the increase in the number of sections from 8 to 10 causes a reduction of cost and time of total production at about 3.125% and 1.25%, whereas if we increase the number of sections from 10 to 12 we obtain a slight diminution of total cost and time.
Finally we notice that the number of optimum passes and the corresponding manufacturing conditions respecting all the limitations of constraints for a total cutting depth equal to 10.16 mm is obtained with the number of sections N equals to 12.

CONCLUSION
In the current study we propose an approach which uses jointly the Genetic Algorithms and the dynamic programming for the multi-criteria optimization of machining conditions.
The optimization of manufacturing process to several passes was resolved in an efficient manner through the use of dynamic programming.Such procedure allowed determining the number of optimal passes for a given total depth of cut.The optimization of each phase was accomplished through the Genetic Algorithms (AG) method, which provides independently the optimum speed and advance for each pass.We have presented the procedure of determination for the number of optimal sections and the number of optimal passes.

Table 2 :
The optimal value of speed and the cut advance obtained by the genetic algorithms method