Plane Thermoelastic Waves in Infinite Half-Space Caused

FACTA UNIVERSITATIS
Series: Mechanical Engineering Vol. 12, N

o
1, 2014, pp. 27 - 36

POSSIBILITIES OF USING THE MONTE CARLO METHOD

FOR SOLVING MACHINING OPTIMIZATION PROBLEMS


UDC: 519.863; 621.7.01

Miloš Madić, Miroslav Radovanović

University of Niš, Faculty of Mechanical Engineering, Serbia

Abstract Companies operating in today's machining environment are focused on

improving their product quality and decreasing manufacturing cost and time. In their

attempts to meet these objectives, the machining processes optimization is of prime

importance. Among the traditional optimization methods, in recent years, modern meta-

heuristic algorithms are being increasingly applied to solving machining optimization

problems. Regardless of numerous capabilities of the Monte Carlo method, its application

for solving machining optimization problems has been given less attention by researchers

and practitioners. The aim of this paper is to investigate the Monte Carlo method

applicability for solving single-objective machining optimization problems and to analyze

its efficiency by comparing the optimization solutions to those obtained by the past

researchers using meta-heuristic algorithms. For this purpose, five machining optimization

case studies taken from the literature are considered and discussed.

Key Words: Machining, Optimization, Monte Carlo Method

1. INTRODUCTION

In recent years, both high resource efficiency and machining processes optimization are

vital for manufacturing companies to gain a competitive advantage and become market

winners. The ultimate goal of machining optimization is to select machining factor values

so that the overall machining performance is enhanced. Determination of optimal

machining parameters is a continuous engineering task whose goals comprise production

costs reduction as well as achievement of the desired product quality [1]. In general, the

selection of optimal machining parameter values for a specific machine tool plays the most

important role in manufacturing, as the process control parameters of a machine tool are not

always precisely understood. Thus, it becomes increasingly difficult to recommend the

optimum values with an enormous variety of expensive materials in the market [2].

Received February 19, 2014 / Accepted March 22, 2014


Corresponding author: Miloš Madić

University of Niš, Faculty of Mechanical Engineering, Department of Production Engineering, Niš, Serbia

E-mail: madic@masfak.ni.ac.rs

Original scientific paper

28 M. MADIĆ, M. RADOVANOVIĆ

In the real production environment it is a common practice to select machining factor

values based on the experience of the machinist (or production planner), machining

handbooks and manufacturers recommendations. As a result, the user attempts to

optimize the cutting operations by trial-and-error every time he needs to setup the

existing equipment for a new different task [3]. The most adverse effect of such a not-

very scientific practice is decreased productivity due to sub-optimal use of machining

capability [4]. The advances in the machining technology and the developments in related

areas (computing, statistics, artificial intelligence, etc.) have led to development of more

sophisticated approaches including data storage and retrieval, expert systems, model-

based approach, modeless approach based on the Taguchi method, etc. However, the non-

availability of the required technological performance equation represents a major

obstacle to the implementation of optimized cutting conditions in practice [4].

The model-based approach, very popular with researchers, integrates experimental,

statistical, mathematical and artificial intelligence tools thus providing a means for better

understanding of machining processes. Using the experimental data, with the help of

regression analysis, artificial neural networks and fuzzy logic, different empirical

equations for prediction of machining performance characteristic can be developed.

Subsequently, the (near) optimal machining parameter values are determined by the

application of an optimization algorithm such as gradient based, non-gradient, heuristic

or meta-heuristic algorithms. In the field of machining process optimization, the current

trend is the application of meta-heuristic algorithms such as genetic algorithm (GA),

simulated annealing (SA), particle swarm optimization (PSO) algorithm, artificial bee

colony algorithm and ant colony optimization algorithm [5].

Meta-heuristic algorithms perform an efficient and comprehensive exploration of the

optimization search space using random the Monte-Carlo search guided by governing

mechanisms which imitate certain strategies taken from nature, social behavior, physical

laws, etc. Despite numerous capabilities of the Monte Carlo method, its application for

solving machining optimization problems has been given less attention by researchers and

practitioners. The present paper has three objectives: (i) to investigate the Monte Carlo

method applicability for solving single-objective machining optimization problems, (ii) to

develop a framework for solving machining optimization problems using the Monte Carlo

method, and (iii) to analyze efficiency of the Monte Carlo method for solving machining

optimization problems by comparing the optimization solutions to those obtained by the

past researchers using meta-heuristic algorithms. For this purpose, five machining

optimization case studies taken from the literature are considered and discussed.

2. MONTE CARLO METHOD

Many numerical problems in science, engineering, finance, and statistics are solved

nowadays by the Monte Carlo methods, that is, by means of random experiments on a

computer [6]. The Monte Carlo is in fact a class of methods now widely used in computer

simulations [7]. The "classical" Monte Carlo is used as an uncertainty analysis of the

deterministic calculation because it yields distribution describing the probability of

alternative possible values about the nominal (designed) point [8]. The idea of the Monte

Carlo calculation is much older than the computer. The name Monte Carlo is relatively

recent, and is connected to famous casinos in Monaco. It was coined by Nicolas Metropolis

Possibilities of Using Monte Carlo Method for Solving Machining Optimization Problems 29

in 1949 under the name of "statistical sampling". Since the pioneer studies in 1940s and

1950s, especially the work by Ulam, von Newmann, and Metropolis, it has been applied

in almost all area of simulations, from the Ising model to financial market, from

molecular dynamics to engineering, and from the routing of the Internet to climate

simulations [7].

Monte Carlo methods have been used for a long time but only in the last few decades,

they have gained the status of fully rounded numerical methods. In order to obtain

reasonably accurate assessment, it is necessary to calculate a large number of special

cases as well as to carry out a respective statistical analysis; that is why an effective

application of the Monte Carlo methods begins with the emergence of fast computers.

At the heart of any Monte Carlo method is a random number generator: a procedure

that produces an infinite stream of random variables that are independent and identically

distributed according to some probability distribution. When this distribution is a uniform

one (i. e. it has equal probability in the interval from 0 to 1), the generator is said to be a

uniform random number generator [6]. Uniform distribution has a wide-ranging

application in various problems in engineering modeling and optimization.

The Monte Carlo is not only used for estimation but also for optimization purposes.

The optimization based on Monte Carlo methods can be useful for solving optimization

problems with many local optima and complicated constraints, possibly involving a mix

of continuous and discrete variables [6]. In order to enhance the accuracy of the method,

the multistage approach may be applied in which the stochastic computations are

repeated by diminishing the region of search after identifying a near optimal solution.

The basic steps in the Monte Carlo method implementation, illustrated in Fig. 1, are

followed for solving machining optimization problems in this paper.

3. CASE STUDIES

To investigate the efficiency of the Monte Carlo method for solving single-objective

machining optimization problems, five conventional machining process research papers

are considered. Although the Monte Carlo method has universal applicability, the

selection of papers is restricted to only those dealing with the explicitly given

mathematical models i.e. mathematical models in terms of polynomial equations, because

optimization solutions can be readily checked and compared.

3.1. Case study 1

Sharma et al. [9] have conducted turning experiments on aluminum 6061 alloy and

metal matrix composites of aluminum. For turning of Al-SiC (5%) the authors develop

the following relationship between surface roughness and turning parameters:

 

dfdvfvdf

vdfvR
SiCAla





7.4000229.00015.071.42541

000001.04.1044300122.07.18

%5
(1)

where v is the cutting speed (m/min), f is the feed rate (mm/rev), and d is the depth of cut (mm).

30 M. MADIĆ, M. RADOVANOVIĆ

The single-objective machining optimization problem is formulated as follows:

Fig. 1 Monte Carlo method flowchart for solving machining optimization problems

(mm) 10.4

(mm/rev) 0.10.05

(m/min) 740228:subject to

),,,( Minimize





dfvfR
a

(2)

In their attempt to obtain minimum surface roughness and corresponding optimal

turning parameter values, the authors have applied PSO algorithm.

3.2. Case study 2

Sanjeev et al. [10] have investigated the turning process of polymeric material

(polytetrafluoroethylene – PTFE, teflon). The authors have developed regression model

for predicting surface roughness in the following form:

dvdffv

dfvR
a





143.0002.0234.0

175.087.0675.0309.0
(3)

Possibilities of Using Monte Carlo Method for Solving Machining Optimization Problems 31

The single-objective machining optimization problem is formulated as follows:

(mm) 5.20.5

(mm/rev) 0.30.1

(m/min) 275150:subject to

),,,( Minimize





dfvfR
a

(4)

In their attempt to obtain minimum surface roughness and corresponding optimal

turning parameter values, the authors have applied GA.

3.3. Case study 3

Saravanakumar et al. [11] have investigated turning process of the Inconel 718. Using

the experimental data, the authors have developed regression equation for prediction of

material removal rate (MRR) in the following form:

vfdfdvd

vfdfv





78805373431417

17499149311213629819158MRR
(5)

The single-objective machining optimization problem is formulated as follows:

(mm) 0.250.1

(mm/rev) 0.250.15

(m/min) 8060:subject to

),,,(MRR Maximize





dfvf

(6)

In their attempt to obtain maximal MRR and corresponding optimal turning parameter

values, the authors have applied GA.

3.4. Case study 4

Bhushan et al. [12] have investigated turning of Al alloy SiC particle composite

material using carbide inserts. On the basis of the experimental results, the authors have

developed the following regression equation for the prediction of surface roughness:

drfddv

rdfvR
a





56484.033125.342419.30000174.0

18753.019915.419694.000324.072412.0

22
(7)

where r is the tool nose radius (mm).

The single-objective machining optimization problem is formulated as follows:

(mm) 0.80.4

(mm) 0.60.2

(mm/rev) 0.250.15

(m/min) 21090:subject to

),,,,( Minimize





rdfvfR
a

(8)

32 M. MADIĆ, M. RADOVANOVIĆ

In order to obtain minimum surface roughness and corresponding optimal turning

parameter values, the authors have applied GA.

3.5. Case study 5

Poornima and Sukumar [13] have investigated turning of martensitic stainless steel.

On the basis of the experimental results, the authors have developed the following

regression equation for the prediction of surface roughness:

fddvvfdf

vdfvR
a





142.200575.007857.002333.080272.6

00002.047976.030442.101518.051539.1

(9)

The authors have formulated the following single-objective machining optimization

problem:

(mm) 0.50.5

(mm/rev) 0.220.15

(m/min) 12080:subject to

),,,( Minimize
a





dfvfR

(10)

Minimum surface roughness and corresponding optimal turning parameter values are

determined by using GA.

4. RESULTS AND DISCUSSION

In previous research studies the machining optimization problems are solved by using

meta-heuristic algorithms such as the GA and PSO. In this section the optimization

solutions obtained by the past researchers are compared to those obtained by applying the

Monte Carlo method. All calculations are accomplished by the proposed optimization

procedure given in Fig. 1 by using Excel spreadsheet package. The generation of random

numbers is done by using function RAND(). In this paper for solving the machining

optimization problems formulated in previous section, a two-stage Monte Carlo approach

is applied.

In the first stage, random numbers for each independent variable (machining

parameter) are generated by considering the interval ranges for each variable.

Subsequently, 5000 estimations of dependent variable (performance characteristic) are

calculated by using the given mathematical model. After ranking all solutions, the best

solution with extreme (minimal or maximal) value of dependent variable along with

corresponding values of independent variables is identified. In the second stage, on the

basis of the analysis of the previously identified best solution, the range for each

independent variable is modified. Subsequently, the stochastic computations are repeated

again for 5000 estimations, and the best solution is recorded.

The comparison of obtained optimization solutions for the case studies is summarized

in Table 1.

Possibilities of Using Monte Carlo Method for Solving Machining Optimization Problems 33

Table 1 Comparison of machining optimization solutions

Case

study
Method

Machining parameters

Objective function

v (m/min) f (mm/rev)
d

(mm)

IHSA
****

740 0.05 0.4 ---

Ra (µm)

0.22936

PSO 233 0.05 0.4 --- 1.2883
*

0.8745
**

Monte Carlo

method

I stage 609.928 0.051 0.421 --- 0.639

II stage 739.367 0.05 0.4 --- 0.2298

GA 158.065 0.164 1.719 ---
Ra (µm)

61.92

Monte Carlo

method

I stage 151.813 0.291 2.494 --- 38.3504

II stage 150.016 0.3 2.5 --- 37.5009

SA
****

80 0.25 0.1 ---
MRR

(mm
3
/min)

2124.275

GA 79.99 0.25 0.1 --- 2122.23

Monte Carlo

method

I stage 79.928 0.25 0.133 --- 2071.87

II stage 80 0.25 0.1 --- 2124.06

GA 207.055 0.151 0.201 1.199

Ra (µm)

1.039
***

1.0650
**

Monte Carlo

method

I stage 209.293 0.187 0.207 1.108 1.1077

II stage 209.968 0.15 0.2 1.2 1.0499

GA 119.93 0.15 0.5 ---
Ra (µm)

0.74

Monte Carlo

method

I stage 119.051 0.151 0.5 --- 0.7424

II stage 119.979 0.15 0.5 --- 0.7315
*
Results reported by Sharma et al. [9]

**
Corrected values

***
Results reported by Bhushan et al. [12]

****
Results reported by Madić et al. [14]; IHSA – improved harmony search algorithm

The analysis of the machining optimization solutions presented in Table 1 indicates

that: (i) the optimization solutions obtained by the Monte Carlo method after first stage

are comparable to those obtained by past researchers using meta-heuristic algorithms

such as GA and PSO, and (ii) solutions obtained by Monte Carlo method after second

stage are better than those obtained by past researchers using meta-heuristic algorithms.

The optimization solutions presented in this paper indicate that few thousand Monte

Carlo computation runs are efficient for solving multi-dimensional and complex machining

single-objective optimization problems. The efficiency of the Monte Carlo method is

assessed by calculating the percentage improvement of the optimization solution for each

case study. The comparisons are graphically illustrated in Fig. 2.

The entire optimization time when using Monte Carlo method consists of the time

needed to formulate machining optimization problem, the time needed to generate

random numbers for each independent variable considering interval ranges, the time for

Monte Carlo computation runs i.e. evaluation of dependent variable values, the time needed

for ranking the optimization solutions and the time needed for the identification of the

best solution.

34 M. MADIĆ, M. RADOVANOVIĆ

-10

-5

1 2 3 4 5

P
e
rc

e
n

ta
g
e
i
m

p
ro

v
e
m

e
n

Case study

1 2 3 4 5

P
e
rc

e
n

ta
g
e
i
m

p
ro

v
e
m

e
n

Case study

a) b)

Fig. 2 Percentage improvement of the optimization solutions: a) results of Monte Carlo

method after I stage, and b) results of Monte Carlo method after II stage

When the entire optimization time is considered, the Monte Carlo method application

for solving single-objective machining optimization problems requires only few minutes.

The salient advantage of the Monte Carlo method application is that it is possible to

obtain a majority of optimization solutions, which can be particularly advantageous in

machining practice considering different machine/tool constraints. Furthermore, the

Monte Carlo based optimization approach requires no expert knowledge, setting of

algorithm parameters and/or defining an initial solution as in the case of using classical

mathematical and meta-heuristic optimization algorithms.

5. CONCLUSION

In this paper, an attempt has been made to investigate the Monte Carlo method

applicability for solving single-objective machining optimization problems. Five single-

objective machining optimization case studies are considered. In order to analyze the

Monte Carlo efficiency, the optimization solutions obtained are compared to those

determined by past researches using meta-heuristic algorithms. On the basis of the

analysis of the obtained results the following conclusions related to the Monte Carlo

capabilities for solving single-objective machining optimization problems can be made:

 Optimization procedure based on the Monte Carlo method consists of only few
steps and it is very easy to implement without the need to write programming code

or use specialized software packages,

 Monte Carlo is a parameter-free universal method in which optimization search
based on random numbers is independent of initial conditions,

 When the computational time is considered, the Monte Carlo method provides an
efficient determination of solutions,

 The application of the Monte Carlo method is well suited for solving machining
optimization problems and the quality of solutions is comparable or even better

than those obtained by meta-heuristics. Using a multiple-stage procedure one

could expect further enhancement of the optimization search,

 With increasing computational runs, the Monte Carlo method can be more
efficient by avoiding being trapped in local minima,

 Monte Carlo method enables determination of a majority of solutions,

Possibilities of Using Monte Carlo Method for Solving Machining Optimization Problems 35

 Monte Carlo method has the capability for solving multi-objective machining
optimization problems thus marking the scope of future research.

On the basis of obtained results this study proposes the wider usage of the Monte

Carlo method for solving machining optimization problems because of its simplicity,

efficiency and wide-ranging capabilities.

Acknowledgement: The paper is a part of the research done within the project TR35034. The

authors would like to thank to the Ministry of Education and Science, Republic of Serbia.

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36 M. MADIĆ, M. RADOVANOVIĆ

MOGUĆNOSTI PRIMENE MONTE CARLO METODE ZA

REŠAVANJE PROBLEMA OPTIMIZACIJE PARAMETARA

OBRADE

U savremenim tržišnim uslovima kompanije su fokusirane na povećanje kvaliteta proizvoda,

smanjenje troškova i vremena izrade. Za postizanje ovih ciljeva optimizacija parametara obrade je

od izuzetnog značaja. Pored klasičnih metoda optimizacije, poslednjih godina za rešavanje problema

optimizacije se sve češće koriste meta-heuristički algoritmi. Uprkos brojnim mogućnostima Monte

Carlo metode, primena ove metode za rešavanje problema optimizacije parametara obtrade nija

dovoljno istražena. Cilj ovog rada je da se istraži mogućnost primene Monte Carlo metode za

rešavanje jednokriterijumskih problema optimizacije parametara obrade. U ovom radu dobijeni

rezultati optimizacije su upoređeni sa rezultatima optimizacije dobijenih primenom raličitih meta-

heurističkih algoritama. U radu su razmatrane pet studije slučaja jednokriterijumske optimizacije

procesa mašinske obrade.

Ključne reči: mašinska obrada, optimizacija, Monte Carlo metod