Optimization of machining processes using pattern search algorithm

Article history: Received October 2 2013 Received in revised format November 7 2013 Accepted January 1


Introduction
Importance for saving costs, maintaining competitiveness in a fierce market and ever-growing demand for high quality machined products, necessitates optimization of machining processes.Machining processes are highly complex, dynamic processes characterized by a number of machining parameters, i.e. input variables and different performance measures (responses), i.e. outputs (Kovačević et al., 2013).The main goal of optimization of machining processes is to determine the optimal values of machining parameters so as to achieve an enhanced machining performance with high dimensional accuracy (Samanta & Chakraborty, 2011).Traditionally, determination of optimal values of machining parameters comprises of mathematical modeling of a machining performance measures and optimization using an optimization method.
Wide spectrum of optimization methods and algorithms has been proposed for solving machining optimization problems (Dixit & Dixit, 2008;Mukherjee & Ray, 2006;Zhang et al., 2006).Some of these include analytical, classical optimization and artificial intelligence methods.In an early work, Gilbert (1950) presented an analytical approach for the selection of optimal cutting speed in a single pass turning process.Armarego and Brown (1969) reported the application of differential calculus for solving unconstrained machining optimization problem.Bhattacharya et al. (1970) applied Lagrange's method for optimization of unit cost in turning process, subject to the constraints of surface roughness and cutting power.Many references presented the application of linear, non-linear, geometric and dynamic programming methods.
Linear programming was used in the early stage of optimization of machining process (Ermer & Patel, 1974).Later on, Tan and Creese (1995) used a sequential method based on linear programming for optimization of multi-pass turning operation.An approach based on the use of integer linear programming was presented by Gupta et al. (1995) for optimization of machining cost.Linear programming methods are fast and reliable (Al-Sumait et al., 2007), but both objective function and constraint equation(s) are linear functions.As machining optimization problems are mostly complex and non-linear in nature, linear programming methods do not provide an adequate answer, or may not be appropriate for many such problems.Multi-modal objective functions and consideration of multiple non-linear objective functions justify the use of non-linear programming methods (Mukherjee and Ray, 2006).Geometric programming was earlier used for solving different machining optimization problems.Ermer (1971), Petropoulos (1973) and Lambert and Walvekar (1978) applied geometric programming for solving constrained machining economics optimization problems.Sönmez et al. (1999) used geometric and dynamic programming for optimization of multi-pass slab milling and face milling for maximum production rate.The geometric programming method could not become popular for two reasons.First, the constraints and objective functions must be expressible in the form of a polynomial and second, as the number of constraints increase, the degree of difficulty in solving a geometric programming problem increases (Dixit & Dixit, 2008).Dynamic programming has been applied for solving sequential and multi-stage optimization problems (Shin & Joo, 1992;Hayers & Davis, 1979;Sekhon, 1982) and goal programming was earlier used to solve multi-objective non-linear machining optimization problems (Sundaram, 1978;Philipson & Ravindran, 1978;El-Gizawy & El-Sayed, 2002).The aforementioned mathematical programming optimization methods are mostly gradient-based, and they possess many limitations in the application for solving machining optimization problems including: (i) inability to deal with integer/discrete input variables (Zhang et al., 2006), (ii) inclined to obtain a local optimal solution (Yildiz, 2009;Rao & Pawar, 2010;Debroy & Chakraborty, 2013) (iii) a judicious choice of an initial starting point in the input space is required (Zhang et al., 2006;Debroy & Chakraborty, 2013), (iv) slow convergence, (v) lack of robustness (Rao & Pawar, 2010) and (vi) inability to handle the overall machining process complexities due to large number of inter-dependent input variables and their stochastic relationships (Markos et al., 1998).
In the past decade, the new trend in the optimization of the machining processes has been based on the use of meta-heuristic algorithms (Zain et al., 2011;Yusup et al., 2013;Bhushan et al., 2012;Savas & Ozay, 2008;Kilickap et al., 2011;Maji & Pratihar, 2011;Rao & Pawar, 2009;Rao & Pawar, 2010;Rao, 2011;Rao & Kalyankar, 2013;Goswami & Chakraborty, 2014).It has been widely reported that these algorithms have the possibility to deal with discontinuous, non-differentiable, and multidimensional machining process models.Furthermore, they do not require the derivative information of the objective function and constraints for the search, rather, they "intelligently" search the optimization space by combining different rules so as to imitate natural phenomena.Lately, the need to tackle more and more complex machining optimization problems and reach a global optimal solution, led to the introduction of hybrid methods which combine constructive properties of several methods, both classical and meta-heuristic algorithms.Yildiz (2009) demonstrated the superiority of the proposed hybrid method by combining immune algorithm with a hill climbing local search algorithm for solving multi-pass turning operation.Also, hybridization of simulated annealing and Hooke-Jeeves algorithm (Chen & Tsai, 1996), genetic algorithm and simulated annealing (Wang et al., 2004), Taguchi's method and genetic algorithm (Yildiz & Ozturk, 2006), etc., proved the effectiveness and efficiency of combined approach for solving machining optimization problems.
Although the popularity of the meta-heuristic algorithms is ever increasing, they are plagued by their own limitations including (Kovačević et al., 2013;Yildiz, 2009): (i) the optimality of the determined solution is impossible to prove, (ii) algorithm parameters settings have a strong influence on the final optimization solution, (iii) there is no universal rule for setting the algorithm parameters and (iv) even expert knowledge in meta-heuristics, systematical selection of the algorithm parameters, as well as understanding of the optimization problem being solved, do not guarantee the optimality of the obtained solution, (v) premature convergence to a local minimum and poor exploitation abilities.Apart from meta-heuristic algorithms, the potential for solving machining optimization problems also have other optimization methods that are conceptually simpler.In recent years, direct search methods have received renewed interest due to new mathematical analysis, their suitability for parallel and distributed computing, and their utility in addressing optimization problems that involve complex computer simulations (Lewis & Torczon, 2011).Direct search methods as one of the earliest numerical optimization methods, formally proposed in the late 1950s and early 1960s, have remained popular with users due to their (Macklem, 2006;Lewis et al., 2000;Lewis & Torczon, 2011): (i) ease of implementation and formulation requiring setting of only few parameters, (ii) flexibility, reliability and practical success in solving a wide range of non-continual, non-differentiable and multimodal optimization problems, (iii) features unique to direct search methods often avoid the pitfalls that can plague more sophisticated approaches, (iv) robustness in locating at least local optimal solutions.Historically direct search methods can be classified into pattern search (PS) methods, simplex methods, and methods with adaptive sets of search directions (Lewis et al., 2000).The development and results of Torczon's multidirectional search (Torczon, 1989), generalized pattern search (Torczon, 1997), generating set search (Kolda et al., 2003) and mesh adaptive direct search (Audet and Dennis, 2006) renewed interest in the application of direct search methods for solving nonlinear optimization problems.Direct search methods neither compute nor approximate derivatives, instead, they work directly with values of the objective function to drive the search for an optimal point (Lewis & Torczon, 2011).They generate search points according to a pattern, around the current point, and accept points, which improve the objective function.Many of the direct search methods are based on surprisingly sound heuristics that fairly recent analysis demonstrates guarantee global convergence behavior analogous to the results known for globalized quasi-Newton techniques (Lewis et al., 2000).
The main objective of this paper is to introduce the use of PS algorithm to the subject of machining optimization, which to the best of the authors' knowledge, has not been previously applied in this field.This paper aims at investigating applicability and performance of conceptually simple PS algorithm for solving single and multi-objective machining optimization problems.The machining optimization application examples considered are taken from scientific resource bases, such as Springer, Elsevier, and Sage.The paper is organized as follows.After introduction, the brief description of the PS algorithm is given in the second section.In the third section, six case studies of machining optimization problems were considered.In each case study optimization solutions obtained by previous researchers using meta-heuristic algorithms and optimization solutions obtained using the PS algorithm were compared and discussed.Findings and observations are summarized in the last section.

Pattern search algorithm
The PS algorithm is characterized by a series of exploratory moves that consider the behavior of the objective function at a pattern of points, all of which lie on a rational lattice (Lewis et al., 2000).The algorithm computes a sequence of points that may or may not approach an optimal point.
The PS algorithm uses a set of vectors {v i }, called a pattern, to determine which points to search at each iteration.The pattern is defined by the number of independent variables of the objective function, N, and the positive basis set.Two commonly used ones are the maximal basis, with 2N vectors (v At each iteration, the PS algorithm searches a set of points, called a mesh, around the current point (the point computed at the previous step of the algorithm) for a point that improves the objective function value.The mesh is formed by (MathWorks, 2012): 1. Generating a set of vectors {d i } by multiplying each pattern vector v i by a scalar Δ m , called the mesh size.2. Adding the current point to the {d i }.
The pattern vector that produces a mesh point is called its direction.After generation of mesh the PS algorithm polls the points in the current mesh by computing their objective function values.If the PS algorithm finds a point in the mesh that improves the objective function value, the new point becomes the current point in the next iteration.In this case the mesh size Δ m is multiplied by 2 (expansion factor).Otherwise, the poll is called unsuccessful, the current point remains in the next iteration and the mesh size Δ m is multiplied by 0.5 (contraction factor).The PS algorithm stops when any of the following conditions occurs (MathWorks, 2012):  The mesh size is less than mesh tolerance. Maximal number of iterations is reached. Total number of objective function evaluations is reached. Time limit is reached. The distance between the point found in two consecutive iterations and the mesh size are both less than a set tolerance. The change in the objective function in two consecutive iterations and the mesh size are both less than function tolerance.
It should be noted that the implementation of the PS algorithm in the Matlab programming environment allows for customization of the PS algorithm by defining polling, searching, and other functions.For a detailed description of the PS algorithm, its variants and other direct search algorithms refer to Lewis et al., (2000), Torczon, (1989Torczon, ( , 1997)), Kolda et al. (2003) and Audet and Dennis, (2006).

Case studies
To investigate the applicability of the PS algorithm for solving machining optimization problems, six research papers dealing with machining optimization were considered.In order to facilitate validation and comparison of obtained optimization solutions this paper considered only mathematical models developed using polynomial equations.For the purpose of optimization, the related m.files for the considered mathematical models were developed in Matlab.In an initial attempt this study was not focused on the analysis of the effects of main control parameters of the PS algorithm on the quality of optimization solutions obtained and convergence speed.Therefore in all case studies considered the PS algorithm was implemented with the following values of main control parameters.
Poll Mesh poll method: maximal basis 2N initial size: 1 complete poll: off expansion factor: 2 polling order: consecutive contraction factor: 0.5 Prior to the application, the PS algorithm takes at least two input arguments, namely the definition of the mathematical model i.e. objective function and a start point -initial solution.For each case study the optimization was attempted starting from three different initial solutions that are: all machining parameter values set on low level (-1) -"PS solution 1", all machining parameter values set on centre level (0) -"PS solution 2" and all machining parameter values set on high level (+1) -"PS solution 3".All computations were run on Intel Core2Duo T5800 with 4 GB RAM.

Single objective machining optimization examples
3.1.1.Abrasive waterjet machining Çaydaş and Hasçalik (2008) investigated the abrasive waterjet machining process through the application of artificial neural networks and regression analysis.Using the obtained experimental data the authors developed mathematical models to predict surface roughness (R a ) using machining parameters of traverse speed (V), waterjet pressure (P), standoff distance (h), abrasive grit size (d) and abrasive flow rate (m).The final mathematical model for the prediction of R a was obtained as: (1) The range of machining parameter values used in experimental process was selected to present the constraints of the optimization problem as given in Eq. ( 2).
Considering the constraints given in Eq. ( 2), the PS algorithm was used to optimize Eq. (1).The obtained optimization solutions and optimization solutions obtained by past researches using metaheuristic algorithms (Zain et al., 2011;Yusup et al., 2013) are given in Table 1.It can be observed from Table 1 that the PS algorithm gives better results than the genetic algorithm (GA) and simulated annealing (SA) obtained previously by (Zain et al., 2011).The optimization results are comparable with the results of ABC algorithm as previously reported by Yusup et al. (2013).The optimization results indicate that PS algorithm, starting from three different initial points, successfully avoided the local minimum entrapment problem.However, it should be noted that in this case study the initial point had great impact on the convergence speed of the PS algorithm (Fig. 1).As can be observed only 66 iterations were needed to find the optimal solution when starting the optimization from lower bound [50 125 1 60 0.5] as initial point, whereas 104 iterations were needed when starting the optimization from upper bound [150 250 4 120 3.5] as initial point.Considering that the surface roughness mathematical model was complex having five independent variables as well interaction and quadratic terms, the computational time less than 2s indicate that the application of PS algorithm represents an efficient alternative for solving machining optimization problems.

Turning process
In an attempt to optimize turning parameters so as to minimize surface roughness, Bhushan et al. ( 2012) presented an integrated approach consisting of regression analysis and GA.On the basis of the experimental results, Bhushan et al. (2012) developed the following mathematical model for the prediction of surface roughness: (3) where A = cutting speed, B = feed rate, C = depth of cut and D = nose radius.The range of machining parameter values used in experimental process was selected to present the constraints of the optimization problem as given in Eq. ( 4).
The optimization solution of the afore-mentioned optimization problem using the PS algorithm and optimization solution obtained Bhushan et al. (2012) are given in Table 2.
Table 2 indicate that the PS algorithm yielded better results than the GA obtained previously by Bhushan et al. (2012).

Turn-milling machining process
Savas and Ozay ( 2008) investigated tangential turn-milling machining process.The study was aimed at determination of optimum machining parameter values at which surface roughness is minimal by using GA.According to the experiment data obtained the following mathematical model for surface roughness was obtained: where N = workpiece speed, n = tool speed, f = feed rate and a = depth of cut.
The constraints for the machining parameters used in optimization are given in Eq. ( 6).
(rev/min) 700 N 300   (6) The optimization solution of the afore-mentioned optimization problem using the PS algorithm and optimization solution obtained by Savas and Ozay (2008) are given in Table 3. Analysis of obtained optimization results given in Table 3 indicates again the efficiency of the PS algorithm in determining optimization solution which is better than the one obtained by the GA.where x 1 = cutting speed, x 2 = feed rate, x 3 = cutting environment.

Drilling process
The constraints for the machining parameters used in optimization are given in Eq. ( 8).
(m/min) 15 x 5 1   (8) The optimization solution of the afore-mentioned optimization problem using the PS algorithm and optimization solution obtained Kilickap et al. (2011) are given in Table 4. Optimization solutions obtained by the PS algorithm are comparable with the optimization solutions obtained by using the GA.It can be also observed that convergence to the optimal point of the PS algorithm was not affected by the selection of initial point.

Multi-objective machining optimization examples
In multi-objective optimization of the machining processes, instead of treating two objective functions (responses) separately, both are to be simultaneously optimized.

Electrical discharge machining process
Maji and Pratihar (2011) modeled input-output relationships of an electrical discharge machining process based on the experimental data (collected according to a central composite design) using regression analysis.Three machining parameters, such as peak current (I p ), pulse-on-time (T on ) and pulse-duty-factor (t), and two process responses, namely, material removal rate (MRR) and surface roughness (SR) were considered in the study.Both MRR and SR were expressed separately, as given in Eq. ( 9) and Eq. ( 10), respectively: (10) Maji and Pratihar (2011) The obtained optimization solution obtained using the PS algorithm and the solution obtained by Maji and Pratihar (2011) by using the binary coded GA are compared in Table 5.
The obtained optimization results indicate that the PS algorithm can be efficiently used for solving multi-objective machining optimization problems formulated based on classic weighted sum method.Fast computational time and avoidance of local optima entrapments confirm the validity on the use of the PS algorithm.From Table 5, it is obvious that the optimal solution obtained by the PS algorithm actually represents the boundary points of the machining parameter values in the covered experimental hyperspace.
3.2.2.Wire electrical discharge machining process Rao and Pawar (2009) investigated the wire electrical discharge machining (WEDM) process.The authors developed mathematical models for correlating different machining parameters and cutting speed (V m ) and surface roughness (R a ).The developed mathematical models by for V m and R a are given by Eqs. ( 12) and ( 13), respectively, whereas Eq. ( 14) gives the surface roughness constraint.
(14) where x 1 = pulse-on time, x 2 = pulse-off time, x 3 = peak current, x 4 = servo feed setting and R per is the permissible value of surface roughness.
The upper and lower bound values for the machining parameters used by Rao and Pawar (2009) are as given as: pulse-on time (µs) = 4 -8; pulse-off time (µs) = 10 -30; peak current (A) = 90 -140; servo feed setting = 30 -50.In order to determine optimal machining parameter values such that permissible surface value of R per = 2 μm is obtained and cutting speed is maximized at the same time, different meta-heuristic algorithms were previously applied (Rao & Pawar, 2009;Rao, 2011;Rao & Kalyankar, 2013) (Table 6).For solving nonlinear constraint problems, as in this case study, the PS algoritm uses augmented Lagrangian approach, in which the bounds and linear constraints are handled separately from nonlinear constraints.A subproblem is formulated by combining the objective function and nonlinear constraint function using the Lagrangian and the penalty parameters.A sequence of such optimization problems are approximately minimized using the PS algorithm such that the linear constraints and bounds are satisfied (MathWorks, 2012).The optimization solutions obtained using the PS algorithm are given in Table 6.When solving nonlinear constrained optimization problems, as in this case study, the selection of starting point resulted in three different optimization solutions.Namely, no feasible solution was found when starting the optimization from lower bound [4 10 90 30] as the initial point.PS optimization led to the maximum V m of 1.3997 mm/min (local maximum) when starting the optimization from upper bound [8 30 140 50] as the initial point.However, the best obtained solution by the PS algorithm was obtained when starting the optimization from [6 20 115 40] as the initial point.It is interesting to note that this solution was found only in four iterations, requiring however, computational time of about 8 s.When comparing the optimization solution obtained by the PS algorithm with the solutions of other meta-heuristic algorithms it can be observed that the PS algorithm solution is comparable to the solution obtained by the particle swarm optimization (PSO) algorithm, and better than solutions obtained by SA, shuffled frog leaping (SFL) algorithm and modified harmony search (MHS) algorithm, all reported by Rao (2011).As can be seen from Table 6, the best optimization solution was obtained by the artificial bee colony (ABC) algorithm (Rao and Pawar, 2009).

Conclusions
This paper has introduced a new approach based on the PS algorithm for solving machining optimization problems.The PS algorithm has been used to solve both the single-and multi-objective optimization problems which had been solved by the past researchers using meta-heuristic algorithms.The conclusions of this research are summarized in the following points:  Convergence of the PS algorithm, i.e. the number of required iterations and computational time greatly depends on the selection of the initial solution.The effectiveness of the PS algorithm appears to rely on how close the initial point is to the optimal point. Unlike meta-heuristic algorithms, the PS algorithm consistently produces the same solutions when the optimization is started from the same initial point. In the case of solving single-objective optimization problems, the optimization solutions obtained by the PS algorithm are better or at least comparable with the optimization solutions obtained by using meta-heuristic algorithms such as GA, SA and ABC algorithm. The PS algorithm can be efficiently used for solving multi-objective machining optimization problems formulated on the basis of the classic weighted sum method. Although the PS algorithm has yielded comparable or better optimization solutions than the optimization solutions obtained by several meta-heuristic algorithms such as PSO, SA, SFL and MHS, when solving nonlinear constrained machining optimization problems, the PS algorithm has faced convergence problems.The comparative analysis of optimization results has indicated that the priority in this case should be given to the ABC algorithm, which has powerful mathematical tools to guide the exploration of optimization space.
In conclusion, it has been found that the PS algorithm is an efficient optimization method and its overall performance has shown that it is well suited for solving machining optimization problems.
Considering that the previous researchers have adjusted the main parameters of GA and ABC algorithm (Yusup et al., 2013;Bhushan et al., 2012;Kilickap et al., 2011;Maji & Pratihar, 2011), while in this paper the PS optimization solutions have been obtained without any adjustments of the main PS algorithm parameters such as mesh size, expansion and contraction factor values, one can conclude that deterministic direct search methods, such as the PS algorithm, have good competitive potential in solving machining optimization problems against stochastic direct search methods such as metaheuristic algorithms.The main scope of future work will be the analysis of the PS algorithm parameters and selection of initial solutions by the use of Taguchi's experimental design technique and the application with comparative analysis of other direct search methods for solving machining optimization problems.Attempts will also be made to investigate the efficiency of a combined optimization approach by integrating deterministic and stochastic direct search methods.
Kilickap et al. (2011) presented a GA based methodology for optimization of drilling parameters considering surface roughness as objective function in drilling of AISI 1045.Mathematical prediction model of the surface roughness was obtained as , (7)

Table 1
Comparison of optimization solutions for abrasive waterjet machining process

Table 2
Comparison of optimization solutions for turning process

Table 3
Comparison of optimization solutions for turn-milling machining process formulated the multi-objective problem considering both MRR and SR as given bellow:

Table 5
Comparison of optimization solutions for electrical discharge machining process

Table 6
Comparison of optimization solutions for wire electrical discharge machining process