ORIGINAL ARTICLE
Modeling and optimization of surface roughness in keyway milling using ANN, genetic algorithm, and particle swarm optimization
Gourhari Ghosh1&Prosun Mandal1&Subhas Chandra Mondal1
Received: 12 March 2017 / Accepted: 15 November 2017 / Published online: 28 November 2017
#Springer-Verlag London Ltd., part of Springer Nature 2017
Abstract
This paper emphasizes on the development of a combined study of surface roughness for modeling and optimization of cutting parameters for keyway milling operation of C40 steel under wet condition. Spindle speed, feed, and depth of cut are considered as input parameters and surface roughness (Ra) is selected as output parameter. Surface roughness model is developed by both artificial neural networks (ANN) and response surface methodology (RSM). ANOVA analysis is performed to determine the effect of process parameters on the response. Back-propagation algorithm based on Levenberg-Marquardt (LM) and gradient descent (GDX) methods is used separately to train the neural network and results obtained from the two methods are compared. It is found that network trained by the LM algorithm gives better result. ANN model (trained by the LM algorithm) is coupled with genetic algorithm (GA) and RS model is further interfaced with the GA and particle swarm optimization (PSO) to optimize the cutting conditions that lead to minimum surface roughness. It is found that RSM coupled with PSO gives better result and the result is validated by confirmation test. Good agreement is observed between the predictedRavalue and experimentalRavalue for RSM-PSO technique.
Keywords Surface roughness . Keyway milling . Artificial neural network . Genetic algorithm . Particle swarm optimization
1 Introduction
In the era of global competition, manufacturers are compelled to find the ways to increase the productivity and to improve the quality of the product. End milling is a widely accepted metal removal process in industry owing to its higher material removal rate and ability to produce good surface quality [1]. It is extensively used to make slots, pockets, and dies in the industries like aerospace and automotive sectors. Keyway is an inevitable part of transmission shaft or hub to couple the rotating machine elements like sprockets, pulleys, gears, and flywheels with the transmission shaft by inserting a key through the keyway. Keyed joint should have the ability to restrict the relative rotational motion and axial motion be- tween the transmission shaft and the rotating elements. The friction between the keyway of shaft or hub and key prevents
the relative motion between shaft and hub [2]. Surface rough- ness has a significant effect on friction. However, a tight tol- erance cannot be accomplished with a very rough surface finish. Keyway should be moderately finished to increase the life of both key and keyway. Generally, surface roughness of keyway should not exceed 6.35μm and it should not also be in the submicron range (below 1μm) to maintain the nec- essary friction between the key and keyway [3].
Surface roughness is a critical parameter to assess the qual- ity of machined surface. It has great influence on surface properties like friction, wear resistance, and fatigue. It depends on the various process parameters and machining conditions.
Due to its nonlinearity, the analytical approach for its model- ing is very difficult. Various theoretical models have been proposed so far which are not sufficient to accommodate the wide range of cutting conditions. Thus, it is needed to develop a tool that can accurately predict the surface roughness of machined surface and also select optimum machining parameters.
Noordin et al. [4] investigated the performance of a multi- layer tungsten carbide tool for turning AISI 1045 steel using response surface methodology (RSM). The input factors (cut- ting speed, feed, and the side cutting edge angle) and
* Gourhari Ghosh
1 Department of Mechanical Engineering, Indian Institute of Engineering Science and Technology, Shibpur, Howrah 711103, India
corresponding responses (cutting force, i.e., the tangential force and surface roughness) were investigated. It was found that the feed is the most significant factor that influences the surface roughness and the tangential force. Tzeng and Chen [5] have applied response surface methodology and back- propagation neural network (BPNN) based on Levenberg- Marquardt (LM) training algorithm integrated with genetic algorithm (GA), separately to optimize the EDM process pa- rameters. It is perceived that BPNN/GA gives better results than the RSM approach. Suresh et al. [6] developed a surface roughness model using response surface methodology (RSM) for machining of mild steel with TiN-coated tungsten carbide cutting tools. The surface roughness model was coupled with genetic algorithms (GA) to find optimum cutting condition.
Prakasvudhisarn et al. [7] have developed model of surface roughness using support vector machines (SVMs) and find optimal cutting parameters by particle swarm optimization (PSO). Deng et al. [8] have developed a BP neural network model to reveal the complex nonlinear relationship between process parameters and responses in camshaft grinding. A tangent transfer function was used to calculate the output of each neuron. Thereafter, GA is used to improve the accuracy based on ANN model and to optimize the process parameters.
H. Oktem [9] has proposed a methodology to develop a sur- face roughness model and optimize the cutting parameters during end milling of AISI 1040 steel with TiAlN solid car- bide tools under wet condition. Genetic algorithm (GA) coupled with tested ANN is used to determine the optimum cutting parameters. Zain et al. [10] have integrated artificial neural network (ANN) and genetic algorithm (GA) techniques to determine the optimal cutting parameters in end milling.
ANN models based on the feed-forward back-propagation algorithm were developed using MATLAB ANN toolbox. It is observed that time of searching for optimal solution was reduced by the integrated ANN-GA approach compared to the conventional GA. Oktem et al. [11] have coupled response surface methodology (RSM) with a developed genetic algo- rithm (GA) to optimize the process parameters in milling of mold surfaces. RSM is used to develop a surface roughness model in terms of cutting parameters like feed, cutting speed, axial depth of cut, and radial depth of cut. RS model is further interfaced with the GA to optimize the cutting conditions for desired surface roughness. Dikshit et al. [12] have integrated genetic algorithm (GA) with response surface methodology (RSM) for optimization of cutting parameters in high speed ball-end milling of Al12014-T6 under dry condition. Tsai et al. [13] have developed artificial neural network (ANN) model based on back-propagation learning algorithm for predicting the surface roughness during end milling process.
It is observed that the two-hidden-layer ANN model has ca- pability to predict theRavalue more accurately and efficiently compared to the one-hidden-layer. Alauddin et al.[14] have presented an approach to develop mathematical models for
tool life in the end milling of steel (190 BHN) by response surface methodology (RSM). Al-Zubaidi et al. [15] have employed newly developed gravitational search algorithm (GSA) to optimize the machining parameters in end milling.
It is perceived that GSA is faster than GA to achieve optimal solution. Jeyakumar et al. [16] have coupled GA with both RSM and ANN model to obtain minimum surface roughness in end milling operation. BP neural network with gradient descent method (GDX) is used to train the ANN network. It is observed that the ANN/GA gives better results than the RSM/GA. Premnath et al. [17] have developed a response surface modal (RSM) to predict the surface roughness during face milling of hybrid composites. RSM was employed to create a mathematical model and adequacy of the model was verified using analysis of variance. Zhong et al. [18] have predicted the surface roughness heightsRaand Rtof turned surfaces using neural networks with seven inputs, namely, tool insert grade, workpiece material, tool nose radius, rake angle, depth of cut, spindle rate, and feed rate. Huang et al. [19] have developed the neural network-based surface roughness Pokayoke (NN-SRPo) system to maintain the surface rough- ness within a desired limit by controlling the machining pa- rameters online. It is found that average difference between predicted and measuredRawas within 5μm. Particle swarm optimization (PSO) algorithm first developed by Kennedy and Eberhart [20] and it is used for optimization of continuous nonlinear functions. Raja and Baskar [21,22] applied PSO technique to determine the optimal machining parameters for minimizing machining time and surface roughness during CNC turning operation of brass, aluminum, copper, and mild steel. Furthermore, PSO was also applied to optimize the cut- ting parameters in face milling for minimizing the surface roughness. It is found that the difference between experimen- tal and predicted surface roughness is very small and accuracy rate 85%. Optimal cutting parameters were successfully found with the help of PSO for certain requirements such as cutting force, surface finish, and cutting tool life [23,24]. Malviya and Pratihar [25] applied PSO for tuning of the neural net- works that were utilized for carrying out both forward and reverse mapping of different metal inert gas welding process- es. Zain et al. [26] have integrated genetic algorithm (GA) and simulated annealing (SA) technique to obtain optimal machin- ing parameters that leads to minimum surface roughness in end milling of Ti-6Al-4V. Integrated SA-GA has an ability to search faster than the conventional SA or GA. Gupta et al. [27] performed multi-objective optimization using PSO and desirability approach search for a set of optimal cutting parameters in turning of titanium alloy under minimum quan- tity lubrication environment. It was observed that PSO gives closer values compared to the results obtained with the desir- ability approach. Tamang et al. [28] coupled the developed ANN model with PSO for optimization of cutting parameters in turning of Inconel 825. The‘purelin’processing function
was employed for the output layer of ANN network. It is found that PSO has better convergent capability with mini- mum number of iterations. Malghan et al. [29] utilized RSM to study the relationship between the input and output re- sponses in face milling of aluminum matrix composites.
Optimization is performed by the desirability approach and the PSO technique. Kumar et al. [30] have developed a surface roughness model using regression analysis in turning. Both GA and PSO are used to optimize the machining parameters and it is observed that GA gives better result. Selaimia et al.
[31] used RSM to make a relationship between input param- eters and output responses in dry face milling of austenitic stainless steel. Desirability function (DF) is used to minimize
surface roughness (Ra) and maximize material removal rate (MRR). It is observed that Rais mostly influenced by feed per tooth and MRR is mostly influenced by both feed per tooth and depth of cut.
From the previous discussions, it is perceived that keyway has some significant applications and roughness is an impor- tant property of the keyway. It is also observed from the liter- ature survey that the surface roughness modeling of keyway in C 40 steel (commonly used as an ordinary transmission shaft material) and optimization of machining parameters are rarely addressed. In this present study, spindle speed (N), feed (f), and depth of cut (d) are considered as process parameters, whereas surface roughness (Ra) is selected as the output Fig. 1 The proposed
methodology
Fig. 2 Artificial neural network architecture
parameter. Surface roughness model is developed by both artificial neural networks (ANN) and response surface meth- odology (RSM). Back-propagation algorithm based on Levenberg-Marquardt (LM) and gradient descent (GDX) methods is used separately to train the neural network and results obtained from the two methods are compared. It is perceived that BPNN with Levenberg-Marquardt (LM) train- ing algorithm is more effective and efficient method than the BPNN with gradient descent (GDX) method. Thereafter, ANN model based on the BPNN trained with LM method is coupled with genetic algorithm (GA) for optimization of the process parameters. Furthermore, RS model is also coupled with both genetic algorithm (GA) and particle swarm optimi- zation (PSO) techniques to obtain optimum cutting parameters that lead to the minimum surface roughness. Results obtained from different optimization techniques are compared and it is found that RS model coupled with PSO (RSM-PSO) gives the minimum surface roughness. Validation experiments are per- formed and a good agreement is observed between the RSM- PSO predicted surface roughness (Ra) value and experimental Ravalue.
2 Proposed methodology
Figure1shows the flowchart of proposed methodology. Full factorial design of experiments method is used to plan the experiments. Thereafter, experiments are carried out accord- ing to the plan and the surface roughness (Ra) is measured for all the samples. The surface roughness (Ra) corresponding to the each parameter setting is tabulated. To study the relation- ship between the surface roughness and the input parameters (i.e., spindle speed, feed, and depth of cut), surface roughness models using both RSM and ANN techniques are developed.
BPNN with Levenberg-Marquardt (LM) and gradient descent method (GDX) methods are used separately to develop two different ANN models. The performances of two ANN
models are compared based on percentage error of predicted Raand experimentalRa. The best ANN model is coupled with the GA for optimizing the machining parameters. The devel- oped RS model is further coupled with GA and PSO.
Thereafter, the results obtained from different optimization techniques are compared. Finally, confirmation experiments are carried out to validate the results obtained from optimiza- tion techniques.
2.1 Modeling using artificial neural network (ANN) ANN is one of the most popular nonlinear mapping systems in artificial intelligence which has the ability to solve many prob- lems including modeling, predicting, and measuring in exper- imental knowledge [5]. Basically, a neural network consists of a number of processing elements linked together via weighted interconnections [8,9]. Common configurations of neural net- works are fully interconnected. Each processing element re- ceives input signals via weighted incoming connections. The processing and learning ability of a neural network largely depends on the strengths of inter-unit connection, or weights.
The training algorithm (or learning) is defined as a procedure that consists of adjusting the weights and biases of a network to minimize selected function of error between the actual and desired outputs [32–34]. The ANN architecture which is used in this paper for modeling and predicting surface roughness in keyway milling is shown in Fig.2.
The accuracy, reliability, and effectiveness of the neural network greatly rely on the number of training and testing data, learning rate, momentum rate, number of hidden layers, and processing function used [35].
In this present study, the back-propagation algorithm (BP algorithm) based on Levenberg-Marquardt (LM) and gradient descent method (GDX) with variable learning and momentum rate is used separately to train the network [35]. Nguyen- Widrow weight initialization algorithm has been applied. BP learning algorithm uses gradient descent method or Levenberg-Marquardt (LM) algorithm to minimize the mean square error between the desired output and the network out- put. A multi-layered ANN based on BP learning algorithm
Table 1 Chemical composition of C40 steel (wt%)
Elements C Si Mn Ni P S Cr Mo
Percentage 0.37–0.44 ≤0.4 0.5–0.8 ≤0.4 ≤0.045 ≤0.045 ≤0.4 ≤0.1
Table 2 Mechanical properties of C40 steel Tensile strength
(MPa)
Upper yield point (MPa)
Elongation (%)
Brinell hardness (BHN)
580 320 6–10 163–211
Fig. 3 Workpiece material
can be effectively created by utilizing the equations in the following (Eq.1) [35].
NETj¼ ∑n
j¼0WijXi ð1Þ
In neural network, each neuron receives total input from all of the neuron in the previous layer. In Eq.1,Wijis the con- nection weight from theith input neuron to thejth hidden neuron,Xiis theith input, andNis the number of inputs to thejth hidden neuron.NETjis the sum of the weighted outputs and transferred into the activation function (F) (Eq.2) which gives the output (OUTj) (Eq.3) of thejth neuron in the next layers [9]. Sigmoid function is chosen as an activation func- tion for the present study.
F NET j
¼ 1
1þeð−NETjÞ ð2Þ
OUTj
¼F NETj
ð3Þ
BP learning algorithm updates the weights and trains the neural network until the mean square error (MSE) converges to a minimum value between the desired output and the net- work output. The network system error is calculated by Eq.4.
MSE¼1 2 ∑M
m¼1 ∑K
k¼1
DESmk−OUTmk
ð Þ2 ð4Þ
whereDESmkandOUTmkare the desired output and the net- work output,Kis the number of output neuron, andMis the overall number of data set [9,10].
The adjustment of the weights can be defined as follows (Eq.5)
ΔWijð Þ ¼n −η* δE
δWijþα*ΔWijðn−1Þ ð5Þ whereηis the learning rate,αis the momentum rate, andn is the iteration.
The outputs associated to a new testing pattern are deter- mined and examined whether the obtained deviations from Table 3 Specification of
cutting tool Manufacturer’s name Miranda Tools Cutting tool material HSS Diameter of cutter (D) 10 mm
Length (L) 72 mm
Height of cutter (H) 25 mm
Helix angle 30°
Table 4 The input process parameters and their levels
Factors Symbols Low level Medium level High level
Spindle speed (rpm) N 180 355 500
Feed rate (mm/min) f 25 80 125
Depth of cut (mm) d .2 .6 1
Fig. 4 Experimental setup for end milling operation
desired values are reasonably small or not. If no, back- propagation should be again considered with revised network by changing the number of neurons, altering learning rate and momentum rate.
2.2 Modeling using response surface methodology (RSM)
Response surface methodology is a collection of math- ematical and statistical techniques that are useful for the modeling and analysis of problems in which a response of interest is influenced by several variables and the objective is to optimize this response [6, 11, and 12].
The RSM is a dynamic and foremost important tool of design of experiment (DOE) wherein the relationship between responses of a process with its input decision variables is mapped to achieve the objective of maximi- zation or minimization of the response properties [6, 36]. The whole concept of response surface involves
dependent variable Y also called the response variable and the several independent variables X1, X2, X3, X4,
………, Xn.
Polynomial curve fitting equations normally exist in first degree and second degree also referred as first-order or the second-order polynomials. The first-order polynomials have the form (Eq.6) [12]
Yu¼β0þβ1X1uþβ2X2uþB3X3uþB4X4u
þ :……… þβkXkuþεu ð6Þ where the termXiurepresents the level of theith factor in the uth experiment. β0,β1, β2,β3, β4…βkare the model parameters. In most of cases, the response surface variables demonstrate some curvature in most of the cutting parameters [14]. Therefore, it is useful to consider the second-order model in this study to evaluate the parametric effects on various response criteria as follows (Eq.7)
YU ¼β0 þ∑Bixi þ∑βiiXi2þ ∑βijXiXj
þ ε;ði¼1……:k and i< jÞ ð7Þ Theβparameters of the polynomials are estimated by the method of least squares. The second-order model helps to understand the second-order effect of each factor separately and the two-way interaction among these factors combined.
2.3 Optimization using genetic algorithm (GA) The GA (genetic algorithm) is a population-based meta- heuristic method for solving optimization problems in the engineering, mathematics, and the other fields [37].
GA is computerized searching and optimization algo- rithm based on Darwin’s evolutionary computation tech- nique which presents the idea of “survival of the fittest”
and “natural selection” [38, 39]. These algorithms gen- erate a population of solutions and make them evolve by encouraging the survival and the reproduction of the solutions which are the most likely to converge toward the optimum.
The GA hopes to converge on the better solution by beginning with a set of potential solution changing them through several generations [40]. The solution of an optimization problem with the GA algorithm begins with a set of potential solution that is known as chro- mosomes. The entire sets of these chromosomes com- prise populations which are randomly selected [9]. The entire set of these chromosomes evolve during several generations or iterations. New generations known as off- spring are generated by utilizing the crossover and mu- tation techniques [5,10]. Crossover involves the process of splitting two chromosomes and then combining one Table 5 Experimental design matrix and surface roughness (Ra)
Ex. no. N(rpm) f(mm/min) d(mm) Ra(μm)
1 180 25 0.2 2.34
2 180 25 0.6 2.62
3 180 25 1 3.90
4 180 80 0.2 4.15
5 180 80 0.6 4.20
6 180 80 1 5.50
7 180 125 0.2 8.85
8 180 125 0.6 9.10
9 180 125 1 10.20
10 355 25 0.2 1.88
11 355 25 0.6 2.24
12 355 25 1 2.33
13 355 80 0.2 3.23
14 355 80 0.6 3.26
15 355 80 1 3.31
16 355 125 0.2 5.42
17 355 125 0.6 8.56
18 355 125 1 9.20
19 500 25 0.2 1.40
20 500 25 0.6 2.01
21 500 25 1 2.13
22 500 80 0.2 2.68
23 500 80 0.6 3.05
24 500 80 1 3.28
25 500 125 0.2 3.98
26 500 125 0.6 4.85
27 500 125 1 5.45
half of each chromosome with the other pair. Mutation involves the process of flipping a chromosome. The
genetic algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm Table 6 Training and testing
error for different network architectures (LM)
Sl.
no
Architecture Learning rate
(η) Momentum rate
(α) MSE
(training)
MSE (testing)
Number of iterations
1 3-3-1 0.1 0.9 0.0623 0.1130 19
2 3-3-1 0.2 0.7 0.1640 0.5281 13
3 3-3-1 0.2 0.8 0.0841 0.1617 16
4 3-3-1 0.1 0.6 0.0648 0.0975 35
5 3-3-1 0.01 0.8 0.0579 0.0643 13
6 3-3-1 0.01 0.5 0.2211 0.3863 38
7 3-3-1 0.02 0.9 0.3301 0.2793 684
8 3-4-1 0.2 0.8 0.1082 0.0905 36
9 3-4-1 0.1 0.9 0.0250 0.0216 24
10 3-4-1 0.1 0.5 0.0259 0.4316 15
11 3-4-1 0.4 0.8 0.0620 0.0809 12
12 3-4-1 0.01 0.6 0.0966 0.1230 14
13 3-4-1 0.02 0.9 0.0337 0.0330 15
14 3-5-1 0.01 0.9 0.0067 0.1143 40
15 3-5-1 0.02 0.6 0.0146 0.0177 16
16 3-5-1 0.1 0.8 0.2565 0.2460 11
17 3-5-1 0.2 0.9 0.0004 0.0006 23
18 3-6-1 0.1 0.8 0.0432 0.0948 14
19 3-6-1 0.02 0.6 0.0179 2.3789 17
20 3-6-1 0.03 0.5 0.0459 0.0972 31
Table 7 Training and testing error for different network architectures (GDX)
Sl.
no
Architecture Learning rate
(η) Momentum rate
(α) MSE
(training)
MSE (testing)
Number of iterations
1 3-3-1 0.1 0.9 0.2180 0.2096 3152
2 3-3-1 0.2 0.7 0.0456 0.0579 10,596
3 3-3-1 0.2 0.8 0.1062 0.0584 3370
4 3-3-1 0.1 0.6 0.0721 0.1246 1827
5 3-3-1 0.01 0.8 0.1209 0.1578 24,141
6 3-3-1 0.01 0.5 0.3077 0.3912 3115
7 3-3-1 0.02 0.9 0.2427 0.2354 3613
8 3-4-1 0.2 0.8 0.1248 0.1737 1658
9 3-4-1 0.1 0.9 0.0647 0.0949 6290
10 3-4-1 0.1 0.5 0.0278 0.0439 5657
11 3-4-1 0.4 0.8 0.1607 0.2007 485
12 3-4-1 0.01 0.6 0.0769 0.1379 49,865
13 3-4-1 0.02 0.9 0.1268 0.4553 2401
14 3-5-1 0.01 0.9 0.0990 0.0610 8591
15 3-5-1 0.02 0.6 0.0359 0.1070 35,304
16 3-5-1 0.1 0.8 0.0290 0.2022 20,936
17 3-5-1 0.2 0.9 0.0093 0.0887 19,143
18 3-6-1 0.1 0.8 0.0228 0.1560 11,721
19 3-6-1 0.02 0.6 0.0785 0.3061 32,935
20 3-6-1 0.03 0.5 0.0160 0.0135 43,453
selects individuals at random from the current popula- tion to be parents and uses them to produce the children f o r t h e n e x t g e n e r a t i o n [11] . O v e r s u c c e s s i v e
generations, the population evolves toward an optimal solution. In order to get an optimal solution, the gener- ated population is evaluated by employing a certain Fig. 5 Variation of MSE with
number of iteration for 3-5-1 neural network withη= 0.2 and α= 0.9 (LM)
Fig. 6 Regression plot for training, validation, and testing for 3-5-1 neural network (LM)
Fig. 7 Variation of MSE with number of iteration for 3-5-1 neural network (GDX)
Fig. 8 Regression plot for training, validation, and testing for 3-5-1 neural network (GDX)
fitness criterion. Primarily, the evaluation process is re- peated until one chromosome with the best fitness criteria is obtained. Then, this best fitness is taken as the optimum solution for the problem [8].
2.4 Optimization using particle swarm optimization techniques (PSO)
In recent years, swarm intelligent-based algorithms have gained a lot of importance and become the research interest [7]. Swarm intelligence is an innovative distrib- uted intelligent concept for solving optimization prob- lems that originally took its motivation from the biolog- ical examples of swarming of bees and flocking of birds [41–43]. PSO is introduced to perform the optimization of continuous nonlinear functions. The swarm is com- posed of volumeless particles with stochastic velocities, each of which represents a feasible solution. The algo- rithm finds the optimal solution through moving the particles in the solution space. In PSO algorithm, the flight trajectory of a particle is influenced by the trajec- tory of neighborhood particles and the flight experience of the particle itself [23]. Each particle is treated as a point in the n-dimensional space. Each particle keeps information of its best coordinates in the problem space.
These coordinates are the best solutions that the particle
has achieved till that time. It is termed as pbest (per- sonal best) [7, 25]. The particle also keeps track of other particles in the neighborhood. So, it is important to check whether there is any other particle in the neighborhood that has better coordinates other than this particle. If there is another particle with better coordi- nates than its coordinates, then that particle’s coordi- nates are termed as local best (lbest); otherwise, the coordinates of the considered particle are taken as the local best (lbest). The particle with the best coordinate values after all the particles has been compared is termed as the global best value (gbest). This global best value gives the best solution of the optimization prob- lem [20]. Every particle has a velocity component and a position component. Say, for ith particle the position component is represented by xi and the velocity compo- nent is represented by vi. Now, this particle will keep on changing its velocity, along with which the position of the particle also changes. The velocity component is represented by vi=vi1, vi2, vi3,..., vin and the position is represented by xi=xi1, xi2,..., xin [24, 44]. The par- ticles change their velocity and position according to the following equation (Eqs. 8 and 9)
vkiþ1¼wvk1þc1r1pi−xki
þc2r2pg−xki
ð8Þ Table 8 Experimental and
predictedRavalue based on ANN model (LM algorithm)
Ex.
no.
Spindle speed (rpm)
Feed rate (mm/min)
Depth of cut (mm)
ExperimentalRa
value (μm) PredictedRa
value (μm) Error percentage (%)
2 180 25 0.6 2.6200 2.6310 0.419847
3 180 25 1 3.9000 3.9072 0.184615
5 180 80 0.6 4.2000 4.2130 0.309524
6 180 80 1 5.5000 5.5355 0.645455
15 355 80 1 3.3100 3.3608 1.534743
21 500 25 1 2.1300 2.1371 0.333333
26 500 125 0.6 4.8500 4.8708 0.428866
Table 9 Experimental and predictedRavalue based on ANN model (GDX method)
Ex.
no.
Spindle speed (rpm)
Feed rate (mm/min)
Depth of cut (mm)
ExperimentalRa
value (μm)
PredictedRa
value (μm)
Error percentage (%)
1 180 25 0.2 2.3400 1.8003 23.0641
3 180 25 1 3.9000 3.9037 0.094872
4 180 80 0.2 4.1500 4.1027 1.13976
14 355 80 1 3.2600 3.3858 3.858896
15 355 80 1 3.3100 2.8249 14.6556
24 500 80 1 3.2800 3.1306 4.55488
26 500 125 0.6 4.8500 5.0821 4.785567
xkþ1i ¼xki þvki ð9Þ wherei= 1, 2, . . .,N,Nis the size of the population,wis the inertia weight,c1andc2are two positive constants, called the cognitive and social parameter, respectively, andr1andr2are random numbers uniformly distributed within the range (0, 1).
The above equations are used to determine the new velocity and position of particle. This type of optimization process is used to search the entire problem space and is thus helpful in solving complex problems.
3 Experimentation
3.1 Workpiece materialC40 mild steel rod (Fig.3) of 100 mm diameter is used for experiment. For current experiment, C40 mild steel was used because this grade of steel offers better forming and bending quality. It is used for applications, where critical bending op- erations are required. C40 steel has very significant role in industry as it is used as shaft materials. Tables1and2show
chemical composition and mechanical properties of C 40 steel, respectively.
3.2 Data collection for the end milling process Keyway milling of C40 steel bar is conducted in Universal milling machine (Bharat Fritz Werner Ltd., Type-UF-2) using HSS end milling cutter as shown in Fig. 4. The measurements of surface roughness were carried out in Mitutoyo Surftest SJ-301 with cutoff length 2.5 mm and number of sampling length 5, stylus type surface texture-measuring instrument. Specification of cutting tool is given in Table 3.
3.3 Design of experiment
In order to determine the influence of control factors of end milling operation, three input parameters were selected such as depth of cut, feed rate, and spindle speed. For each factor, three levels, high, medium, and low, are considered as shown in Table4.
Table 10 Experimental and predictedRavalue based on RSM model
Ex.
no.
Spindle speed (rpm)
Feed rate (mm/min)
Depth of cut (mm)
ExperimentalRa
value (μm) PredictedRa
value (μm) Errors (%)
2 180 25 0.6 2.6200 2.7102 3.44
3 180 25 1 3.9000 3.1211 19.97
5 180 80 0.6 4.2000 5.0321 19.81
6 180 80 1 5.5000 5.9103 7.47
15 355 80 1 3.3100 4.5423 37.25
21 500 25 1 2.1300 2.9214 37.08
26 500 125 0.6 4.8500 5.1201 5.56
Table 12 Analysis of variance (ANOVA) forRa
Source DOF Sum of squares Mean square Fvalue Pvalue
Model 9 163.548 18.172 34.79 0.000
N 1 25.701 25.701 49.20 0.000
f 1 113.499 113.499 217.28 0.000
d 1 7.012 7.012 13.42 0.002
N2 1 0.028 0.028 0.05 0.039
f2 1 12.111 12.111 23.19 0.000
d2 1 0.006 0.006 0.01 0.019
N*f 1 8.514 8.514 16.30 0.001
N*d 1 0.165 0.165 0.32 0.032
f*d 1 1.127 1.127 2.16 0.029
Error 17 8.880 0.522
Total 26 172.428 Table 11 Regression coefficients in keyway milling
Term Coef SE Coef Tvalue Pvalue
Constant 3.470 0.372 9.33 0.000
N −1.196 0.170 −7.01 0.000
f 2.513 0.170 14.74 0.000
d 0.625 0.171 3.66 0.002
N2 −0.069 0.298 −0.23 0.009
f2 1.437 0.299 4.82 0.000
d2 −0.031 0.295 −0.10 0.008
N*f −0.840 0.208 −4.04 0.001
N*d −0.117 0.208 −0.56 0.030
f*d 0.306 0.208 1.47 0.046
R2= 94.85%,R2(adj) = 92.12%,R2 (pred) = 95.28%
Here, three input parameters (P) and three levels (L) are considered. So numbers of runs are required for a full factorial analysis isN=LP= 33= 27. Experimental design matrix and corresponding response that is surface roughness (Ra) is pre- sented in Table5. The experiments are repeated three times and surface roughness is measured four times for each sample.
Thereafter, average of all the surface roughness (Ra) readings is reported.
4 Results and discussion
4.1 ANN resultsIn the present problem, the network consisted of three input neurons corresponding to spindle speed, feed, and depth of cut and one output neurons corresponding to surface roughness (Ra). The network is trained by two different training algo- rithms viz. Levenberg-Marquardt (LM) and gradient descent method (GDX) with variable learning rate and momentum coefficient. The number of hidden layer, learning rate (η), and momentum coefficient (α) is decided by trial and error method to minimize the mean square error. Table6shows the performances of different networks architecture with different Fig. 9 Normal probability plot of
residual
Fig. 10 Surface plot of surface roughness vs depth of cut and spindle speed
learning rate (η) and momentum coefficient (α) trained by Levenberg-Marquardt (LM) algorithm. The performances for the same network architecture with the same learning rate (η) and momentum coefficient (α) and trained by gradient descent method are presented in Table7.
The training process adjusts the weight of each neuron to an appropriate value. Out of 27 experimental data sets, 20 have been selected at random for training the network, this 20 experiment also used for validation purpose and the re- maining 7 are used for the testing for both the cases (Table6 and Table7). Different combinations of learning rate (η), and momentum coefficient (α) and number of hidden layer have been tried.
Depending on the mean square error, optimum net- work architecture has been selected. Table 6 shows the mean square error (MSE) for training and testing is minimum for the architecture 3-5-1 with learning rate (η) = 0.2 and momentum coefficient α= 0.9. Hence, it is selected as optimum architecture of the network.
Figure 5 shows that variation of mean square error (MSE) for training, testing and validation with iteration.
In that graph, the best point is selected based on the iteration at which the validation performance is reached to minimum value. Graph (Fig. 5) shows that the desire goal for 3-5-1 architecture is achieved at 23 iterations.
The network is validated by regression plots which show the relationship between the outputs of the network and the targets. This regression plot is analyzed for training, valida- tion, and testing pattern. If the training were perfect, the net- work outputs and the targets would be exactly equal, but the relationship is rarely perfect in practice. When theRvalue is greater than 0.9 that indicates the experimental and predicted surface roughness values are much closer. The regression plot for training, testing, and validation is shown in Fig.6. It is observed from the regression analysis that correlation coeffi- cients (R) for training, validation, and testing are 0.99997, 0.9996, and 0.99992, respectively.
Fig. 11 Surface plot of surface roughness vs feed and spindle speed
Fig. 12 Surface plot of surface roughness vs feed and depth of cut
Furthermore, when networks are trained using gradient de- scent method, the architecture 3-5-1 with learning rate (η) = 0.2 and momentum coefficientα= 0.9 gives the minimum MSE (Table7) and it is selected as the best network architec- ture. The performance graph (Fig.7) shows that desire goal for the architecture 3-5-1 is reached at 19137 iterations.
The regression plot for training, testing, and validation for the architecture 3-5-1 with learning rate (η) = 0.2 and momen- tum coefficientα= 0.9 is shown in Fig.8. The correlation coefficients (R) (Fig. 8) for training, validation, and testing are 0.99938, 0.99844, and 0.9835, respectively.
A comparative study has been made between the best networks of two different training algorithms such as Levenberg-Marquardt and gradient descent method.
Though in both the cases, optimum architecture is found to be 3-5-1 with learning rate (η) = 0.2 and momentum coefficient α= 0.9 but higher regression coefficient (R) is achieved using Levenberg-Marquardt training algo- rithm. Number of iteration is much lower for Levenberg- Marquardt training algorithm compared to gradient de- scent training method. So, Levenberg-Marquardt (LM) training algorithm reduces the time of learning, and it also increases the rate of searching. The predicted Ra values and percentage of errors in prediction, based on LM algo- rithm and GDX method, are presented in Table 8 and Table 9, respectively. It is perceived that the percentage
of errors in prediction of Rabased on LM method is very less compare to GDX method. Hence, it can be concluded that Levenberg-Marquardt (LM) algorithm is more effi- cient and effective than the gradient descent method.
4.2 Modeling and optimization using RSM
Response surface method (RSM) adopts both mathemat- ical and statistical techniques which are useful for the modeling and analysis of problems in which a response of interest is influenced by several variables. The devel- oped response surface model is able to predict surface roughness over a wide range of end milling factors.
RSM attempts to analyze the influence of the indepen- dent variables (input variable) on a specific dependent variable (response).The mathematical model commonly used for the machining response Ra (surface roughness) is represented as Eq. 10.
Ra¼f N;ð f;dÞ þε ð10Þ
where N, f, and d are spindle speed, feed, and depth of cut, respectively, and ε is the error which is normally distributed about the observed machining response Ra. Using the experimental data for the surface roughness by MINITAB 17 software the second-order response Table 13 Combination of GA
parameter rates leading to the optimal solution
Population size Elite count Crossover rate Mutation rate Crossover function
80 4 1 0.8 2 point
Fig. 13 Results of the MATLAB optimization toolbox
function was determined below (Eq. 11).
Ra¼1:80þ0:00334 N−0:0095 f þ1:27d−0:000003N2
þ0:000575 f2−0:19 d2−0:000105 N*f−0:00183 N*d þ0:0153 f*d
ð11Þ On putting the experimental values of spindle speed, feed and depth of cut for some selected experimental run in Eq.11, the responses can be calculated. Comparison between exper- imental and predictedRavalue based on RSM model is pre- sented in Table10.
The results of the regression analysis are represented in Table11. When R2approaches to unity, it indicates a good
correlation between the experimental and the predicted values.
It is found that theR2value is 0.95. Hence, it can be concluded that the proposed response surface model is adequate to ex- press the real keyway milling process.
Furthermore, ANOVA analysis was performed to deter- mine the effect of machining parameters on surface roughness and the results are presented in Table12. The ANOVA test was performed at a significance level of 5%, i.e., confidence level of 95%. SincePvalue given in Table12is less than 0.05, it can be concluded that the developed model is significant.
The assumption of normality is checked by using normal probability plot of the residuals and the same is shown in Fig.9. From Fig.9, it is conceived that the proposed surface roughness model follows normal distribution.
By analyzing the 3D surface plots, the optimum machining parameters which lead to minimum surface roughness can be obtained. In each plot, two cutting parameters are varied while Fig. 14 Plot functions of the best
fitness for GA
Fig. 15 Plot functions of the best fitness for PSO
the third one is held at its mid value. Figure10shows the effect of depth of cut (d) and spindle speed (N) on surface roughness.
It is noticed that minimum surface roughness can be achieved when depth of cut is 0.31 mm and spindle speed is 486 rpm. It is also observed that the effect of spindle speed on the surface roughness is greater than the depth of cut.
Figure11shows the effect of feed (f) and spindle speed (N) on surface roughness. It is observed that minimum surface roughness is achieved when feed is 46 mm/min and spindle speed is 486 rpm. It is also observed that the effect of feed on surface roughness is more significant than the spindle speed.
By analyzing Figs.10and11, it is perceived that the surface roughness will be lowest when spindle speed is 486 rpm.
Figure12shows the effect of feed (f) and depth of cut (d) on surface roughness. It is perceived that the minimum surface roughness can be achieved when feed is 46 mm/min and depth of cut 0.31 mm. It is also found that the feed has a significant effect on surface roughness compared to the depth of cut. By analyzing Figs. 11 and 12, it can be concluded that minimum surface roughness can be accomplished when feed is 46 mm/min.
By analyzing three different 3D surface curves, it can be concluded that the minimum surface roughness (2.1779μm)
can be achieved when spindle speed, feed, and depth of cut are 486 rpm, 46 mm/min, and 0.31 mm, respectively.
It is also perceived that the surface roughness increases with the increase of feed rate while an increase in cutting speed decreases the surface roughness. The theoretical surface roughness (Ra) for end milling process can be calculated using the following equation (Eq.12) [45,46]
Ra¼ f2t
32ðRftnt=πÞ ð12Þ
whereftis the feed per tooth (mm/tooth),ntis the number of tooth on the cutter,Ris the radius of the cutter, and the +ve sign is for up milling and–ve sign is for down milling. Hence, it can be concluded from the above equation that the rough- ness will increase if feed increases. Besides, when the cutting speed increases, the cutting temperature also increases. As a consequence, material becomes softer which enhances the cutting performance that leads to the reduction of surface roughness [47]. It is noticed that the effect of depth of cut on the surface roughness is comparatively small. As the depth of cut increases, the contact length, the thickness of uncut mate- rial, and cutting force also increase. Though, the frictional forces associated to a unit contact length at the tool cutting edge and the chip thickness will not change significantly [48].
Hence, depth of cut has small effect on the roughness. The Table 14 Optimal cutting
condition results of PSO Optimal cutting conditions Minimum fitness function
Spindle speed (rpm) Feed (mm/min) Depth of cut (mm) Surface roughness (μm)
500 50.292 0.2 1.2735μm
Fig. 16 Plot functions of the best fitness for GA
same trend is also reported by the previous researchers [49, 50].
4.3 Results obtained from GA and PSO 4.3.1 RSM coupled with GA (RSM-GA)
The aim of the optimization process in this study is to deter- mine the optimal values of decision variables that contribute to the minimum value of surface roughness. To formulate the optimization problem, the response surface model which is proposed in Eq. (11) is taken as the objective function (fitness function). The fitness function (objective function) can be expressed as follows (Eq.11):
Ra¼1:80þ0:00334 N−0:0095 f þ1:27 d−0:000003 N2
þ0:000575 f2−0:19 d2−0:000105 N*f−0:00183 N*d þ0:0153 f*d
ð11Þ The minimization of the fitness function value of Eq. (11) is subjected to the boundaries (limitations) of cutting condition values. The range of values of experimental cutting conditions is used as the limitations of the optimization solution and is presented as follows (Eq.13(a–c)):
180rpm≤N≤500rpm ð13aÞ 25mm=min≤f≤125mm=min ð13bÞ
0:2mm≤d≤1mm ð13cÞ
Basically, obtaining the best optimal results depends on some criteria. The initial population size, the type of selection function, the crossover rate, and the mutation rate have signif- icant influence on optimal results. The value or parameter setting for these criteria is made by the process of trial and error for obtaining the most optimal result.
By using the MATLAB 7.10.0 (R 2010a) optimization toolbox, this study has tried several combinations of the set values for cutting conditions in order to present the best opti- mal results. The best combination of the parameters applied that leads to the minimum values of the fitness function is shown in Table13.
By using the fitness function formulated in Eq. (11), the limitations of cutting conditions formulated in Eq. (13(a-c)) and the GA parameters given in Table13, the results from the MATLAB optimization toolbox are obtained and it is displayed in Figs.13and14.
From Fig. 13, it is observed that the minimum surface roughness value is 2.1655μm. The set values of cutting con- ditions that lead to the minimum surface roughness value are 486.245 rpm for spindle speed, 47.847 mm/min for feed, and 0.306 mm for depth of cut. It is also indicated that the optimal solution is obtained at the 51st generation (iteration) of the GA algorithm. The plot functions (Fig.14) indicate that the both mean and best fitness values are 2.1655μm.
4.3.2 RSM coupled with PSO (RSM-PSO)
The RSM based fitness function (Eq. 11) which is pre- viously used for finding optimal cutting parameters by GA, it is further used for finding optimal cutting Table 15 Optimal parameters
setting and predictedRa Optimization technique
Optimal cutting conditions Minimum fitness
function Spindle speed (N)
(rpm)
Feed (f) (mm/
min)
Depth of cut (d) (mm)
Surface roughness (Ra) (μm)
RSM 486 46 0.31 2.1779
RSM-GA 486.245 47.847 0.306 2.1655
ANN-GA 488.94 82.006 1 1.5479
RSM-PSO 500 50.292 0.2 1.2735
Fig. 17 Results of the MATLAB optimization toolbox
parameters with the same boundary cutting conditions using particle swarm optimization technique. In this case, a number of iteration and population size are se- lected as 100 and 80, respectively. Figure 15 shows the plot functions of best fitness for PSO. The plot func- tions (Fig. 15) indicate that the both mean and best fitness values are 1.2735 μm. Optimal parameters set- tings and corresponding surface roughness (Ra) is repre- sented in the Table 14.
4.3.3 ANN coupled with GA (ANN-GA)
In previous discussions, 3-5-1 ANN architecture with Levenberg-Marquardt (LM) training algorithm was found to be best and efficient. An input and output relationship is de- veloped by applying LM algorithm has been given as follows (Eq.14) [10,51]:
Ra¼ ∑5
j¼1 purelin LWj;1* ∑3
i¼1∑5
j¼1tansig X i*IWi;jþbj!
" #
þa
( )
ð14Þ whereRarepresents surface roughness, purelin gives linear relationship between the input and the output, tansig a hyper- bolic tangent sigmoid transfer function,IWis the first layer weight,LWis the second layer weight,bis the first layer bias, ais the second layer bias, andXrepresents input variables.
Using weight and bias for 3-5-1 architecture, in Eq. (14), an algebraic form of equation was developed. This equation used as fitness function of GA for optimization of keyway milling process parameters to minimize the surface roughness in the same boundary cutting conditions.
Figures16and17show the plot functions of the best fit- ness for GA and the MATLAB optimization toolbox, respec- tively. It is found (Fig.17) that the minimum surface rough- ness (Ra= 1.548 μm) can be accomplished when spindle speed, feed, and depth of cut are 488.99 rpm, 82.006 mm/
min, and 1 mm, respectively.
From the previous discussions, it can be stated that to ob- tain optimal parameter setting that leads to minimum surface roughness in keyway milling, different optimization tech- niques are utilized. To compare the results obtained from dif- ferent optimization techniques, all the optimal parameters
settings and corresponding responses are presented in the Table15.
From Table15, it is notcied that RSM-PSO gives minimum Ra value compared to other proposed methodologies.
Validation experiments are performed based on the available machining parameters values (i.e.,N= 500 rpm,f= 50 and 80 mm/min, and d= 0.2, 0.3, and 1 mm) which are nearest to the optimal parameter settings.
Table16presents the results of validation experiments and the percentage errors of predicted Ra and experimentalRa
through the confirmation tests. It is observed that percentage errror in case of RSM-PSO technique is lesser compared to other techniques. Hence, it can be concluded that RSM-PSO is an efficient and effective technique for optimization of the cutting parameters in keyway milling.
5 Conclusions
From the preceding discussions, the following conclusion may be drawn.
& Back-propagation neural network (BPNN) with
Levenberg-Marquardt (LM) algorithm increases the searching rate which makes it faster than BPNN with gra- dient descent (GDX) method. The percentage of errors in predicting surface roughness (Ra) of keyway based on LM method is lesser than the GDX method. Hence, it can be concluded BPNN with LM algorithm is an efficient and effective technique for surface roughness modeling of keyway.
& Network architecture 3-5-1 with learning rate (η) = 0.2 and
momentum coefficientα= 0.9 for both the algorithms is found as the best architecture on the basis of minimum mean square error (MSE) and this architecture is validated by regression analysis.
& From the 3D surface plots and ANOVA analysis, it is
perceived that the feed has significant effect on surface roughness followed by spindle speed. Depth cut has small effect on roughness. Better surface finish can be obtained by a combination of high speed, low feed, and low depth of cut.
& The results obtained from different optimaization tech-
niques are compared and it is found that the RSM coupled Table 16 Results of validation
experiments Optimaization
technique
Spindle speed (rpm)
Feed (mm/
min)
Depth of cut (mm)
PredictedRa
(μm) Experimental
Ra(μm) Error
(%)
RSM 500 50 0.3 2.1779 1.69 28.86
RSM-GA 500 50 0.3 2.1655 1.69 28.86
ANN-GA 500 80 1 1.5479 3.33 53.51
RSM-PSO 500 50 0.2 1.2735 1.61 20.90
