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Determination of laser cutting process conditions using the preference selection index method
Optics & Laser Technology 89 (2017) 214–220
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Determination of laser cutting process conditions using the preference selection index method
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Miloš Madića, , Jurgita Antuchevicieneb, Miroslav Radovanovića, Dušan Petkovića a b
Faculty of Mechanical Engineering, University of Niš, A. Medvedeva 14, 18000 Niš, Serbia Vilnius Gediminas Technical University, Sauletekio av. 11, LT-10223 Vilnius, Lithuania
A R T I C L E I N F O
A BS T RAC T
Keywords: Laser cutting Optimization MCDM Cut quality Preference selection index method
Determination of adequate parameter settings for improvement of multiple quality and productivity characteristics at the same time is of great practical importance in laser cutting. This paper discusses the application of the preference selection index (PSI) method for discrete optimization of the CO2 laser cutting of stainless steel. The main motivation for application of the PSI method is that it represents an almost unexplored multi-criteria decision making (MCDM) method, and moreover, this method does not require assessment of the considered criteria relative significances. After reviewing and comparing the existing approaches for determination of laser cutting parameter settings, the application of the PSI method was explained in detail. Experiment realization was conducted by using Taguchi's L27 orthogonal array. Roughness of the cut surface, heat affected zone (HAZ), kerf width and material removal rate (MRR) were considered as optimization criteria. The proposed methodology is found to be very useful in real manufacturing environment since it involves simple calculations which are easy to understand and implement. However, while applying the PSI method it was observed that it can not be useful in situations where there exist a large number of alternatives which have attribute values (performances) very close to those which are preferred.
1. Introduction Laser cutting is one of the industry leading technologies for cutting a wide variety of materials. Compared to other alternative cutting technologies, laser cutting offers significant advantages and possibilities such as ability to cut complex geometries with tight tolerances, high cutting speeds, i.e. increased productivity, localized heat affected zone (HAZ), high quality, ease of automation, etc. [1–3]. Although initial capital investments of laser cutting technology are high, operational costs are low making this technology being economically competitive and cost effective. In order to take full advantages and benefits that laser cutting technology offers one needs to carefully consider determination and selection of adequate process conditions, i.e. particular combination of the main parameter values. In general, the selection of parameter settings is mainly dependent on the composition of the workpiece material, workpiece thickness and desired performance characteristics related to cost, quality and productivity. However, the main difficulty is the fact that optimal combination of laser cutting parameter values for one performance characteristic is not even near optimal for other performance characteristics [4–7]. Therefore, determination of laser ⁎
cutting process conditions for multi-performance (multi-criteria) optimization is of prime importance. In the open literature and manufacturing practice one can identify four main approaches for determining of the most suitable laser cutting conditions for a given application: trial and error method, Taguchi method, continual optimization and discrete optimization. Trial and error method is one the most common approaches in real manufacturing environment. It assumes that process planners and engineers use acquired experience and recommendations from handbooks for selecting laser cutting parameter values. This subjective approach leads to the fact that selected values of laser cutting parameters vary from one case to another and are preferably conservative. Although this approach may be sufficient in most applications, it denies the possibility of using better (optimized) cutting conditions and ultimately it does not provide a basis for the full utilization of the laser cutting machine. It can be argued that, due to complexity and stochastic nature of laser cutting, even highly skilled process planners and engineers can hardly determine laser cutting process conditions with respect to several criteria. Due to its simplicity and ease of implementation the application of the Taguchi method is the second most common approach in real
Corresponding author. E-mail addresses: [email protected] (M. Madić), [email protected] (J. Antucheviciene), [email protected] (M. Radovanović), [email protected] (D. Petković). http://dx.doi.org/10.1016/j.optlastec.2016.10.005 Received 24 August 2016; Accepted 24 October 2016 0030-3992/ © 2016 Elsevier Ltd. All rights reserved.
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manufacturing environment. The Taguchi method is a well-known, unique and powerful method for product/process quality improvement. In laser cutting, application of this method allows for identification of near optimal laser cutting process conditions making the process insensitive to noise factors such as environmental temperature, humidity or dust. High popularity of this method is due to fact that this approach does not require development of any mathematical model, thus can be readily applied in real manufacturing environment [4– 6,8,9]. However, on the other hand, this approach permits only discrete optimization and if one need to consider multiple objectives, application of additional methods such as utility method, principal component analysis or principal component analysis is inevitable. Determination of laser cutting process conditions based on continual optimization represents higher level approach. It consists of development of an empirical model of a laser cutting performance characteristic and application of an optimization method. Empirical mathematical models establish relationships between inputs (cutting parameters) and outputs (performance characteristics). To this aim regression analysis, artificial neural networks and genetic programming are predominantly being used [3,7,10–13]. Once developed, these mathematical models are used as objective functions which are optimized using an optimization method. Since laser cutting process optimization problems are complex, highly non-linear and multidimensional, meta-heuristic optimization methods have become a preferred trend for solving these types of optimization problems [2,7,12,14–16]. Integration of empirical models and optimization methods allows for continuous single and multi-criteria optimization of laser cutting process. Namely, laser cutting parameter values (independent variables) are defined in continual or integer domain and the goal is to determine the best solution (laser cutting process condition) which satisfies all previously set constraints and ranges of independent variables. Although providing better optimization solutions this approach is more time and computationally expensive and requires higher knowledge levels from design of experiments, mathematical model development, optimization theory as well as metaheuristics. Finally, the problem of determining of laser cutting process conditions can be viewed as a multi-criteria decision making (MCDM) problem in which a particular cutting conditions represents an alternative while performance characteristics represent criteria upon which alternatives are assessed [17]. Although there is a number of mathematically relatively simple MCDM methods for assessment and ranking of alternatives, determination of laser cutting process conditions by using MCDM methods still do not have wide application in practice. Given that there is a finite number of pre-known alternatives (laser cutting process conditions) this type of multi-criteria optimization problems are referred as discrete. The previously discussed four main approaches are predominantly used in manufacturing practice for determination of laser cutting process conditions. In practice the selection of a given approach is dependent on the particular application. Thus, if there is a need for large batch laser cutting on expensive work materials, the application of continual optimization may the right choice. On the other hand, in situations where waste and possible post-processing imply negligible financial losses, one can select other more simple approaches. Each approach has some advantages, disadvantages and limitations. Although does not provide even near optimal laser cutting process conditions, trial and error method can provide satisfactory results without the need of domain expert knowledge. On the other hand, formulation and solving of multi-criteria continual optimization problems can provide optimal laser cutting process conditions, however, this approach requires a considerable knowledge level of laser cutting, DOE, mathematical modeling, optimization and AI methods. Finally, if time is a limiting factor one can consider discrete optimization approach, whether Taguchi or MCDM methods. Comparison of existing approaches for determination of laser cutting process conditions with
Table 1 Comparison of the existing approaches for determination of laser cutting process conditions.
Computationally expense Implementation time Quality of determined solution Practical suitability Requirements of particular knowledge Possibility to handle a number of criteria Requirements of using specialized software
Trial and error
Taguchi method
Continual optimization
Discrete optimization
–
small
high
small
permanent
fast
average
fast
low
good
high
good
high –
very high some
very high high
average some
none
good
good
very good
–
none
yes
none
respect to different criteria is given in Table 1.
2. Application methodology The application methodology for determination of laser cutting process conditions with the use of the PSI method can be summarized in the six following steps: problem definition, pre-analysis, laser cutting experimental investigation, data acquisition, formulation of decision matrix and application of the PSI method. Problem definition implies determination of the objectives (customer requirements and manufacturers preferences) regarding desired cut quality, productivity and cost. The goal of pre-analysis step is identification of main laser cutting parameters that predominantly influence the objective functions, i.e. laser cutting performance characteristics (surface roughness, kerf width, HAZ, MRR, etc.). Therefore, the laser cutting performance characteristics can be regarded as criteria for evaluation of laser cuts. To this aim in this stage one can perform one factor at a time (OFAT) experimental trials or conduct screening experimental designs such as Plackett-Burman designs. Once the main laser cutting parameters are singled out, experimental trials with different laser cutting parameter values combinations are to be carried out. The laser cutting parameters are to be varied in a range considering manufacturing practices so that wider experimental hyperspace is covered while ensuring that full laser cut is obtained in each combination. To ensure minimal resource use and time saving experimentation one could apply Taguchi's orthogonal arrays, Box-Behnken designs, composite designs, fractional factorial designs, etc. Data acquisition refers to measurement/calculation of quality and performance characteristics that are obtained in each experimental trial, i.e. for each laser cutting parameter values combination. After collection of experimental data one needs to formulate decision matrix based on experimental data for each experimental trial, i.e. combination of laser cutting parameter values with respect to all considered criteria (performance characteristics). Each row in the decision matrix represents one alternative (specific laser cutting process condition), and each column represents one criterion. The final step is the application of the PSI method for deriving the decision rule upon which the most suitable laser cutting process condition can be determined. The goal of the application of any MCDM method is to determine aggregate function, so-called decision 215
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4. PSI method
Table 2 Levels of variation for input variables. Input variable
Level 1
Level 2
Level 3
Cutting speed, v [m/min] Assist gas pressure, p [MPa] Laser power, P [kW] Focal position, f [mm]
2 0.9 1.6 −2.5
2.5 1.05 1.8 −1.5
3 1.2 2 −0.5
The PSI method was proposed by Maniya and Bhatt [18] in 2010 for solving material selection MCDM problems. In 2014, Vahdani et al. [19] proposed interval valued fuzzy PSI method in order to solve complex decision problems under uncertainty. Unlike the most MCDM methods, the PSI method does not require determination of the relative importance of the criteria, and therefore it is not necessary to determine criteria weights. Thus, the method is particularly useful in cases where a conflict in deciding the relative importance among criteria appears [20]. Actually, the PSI method determines criteria weights only by using information provided in the decision matrix, i.e. it uses objective approach to determine criteria weights like standard deviation or entropy method. The application methodology of the PSI method for solving MCDM problems includes several steps [18–20]:
rule, which shows the overall assessment of alternatives using data from the decision matrix as well as considering the preferences of the decision maker regarding relative significances of considered criteria. Based on aggregate function values of alternatives it is possible to obtain complete ranking of alternatives. 3. Experimental details
Step 1: Identification of the relevant criteria for evaluation of the alternatives. Step 2: Development of the initial decision matrix, X:
The CO2 laser cutting experiment was carried out using ByVention 3015 CO2 laser cutting machine. The workpiece material was AISI 304 stainless steel and the sheet thickness was 3 mm. Nitrogen with purity of 99.95% was used as assist gas in experimentation and it was supplied via conical nozzle with 2 mm diameter. The laser beam was focused using lens with focal length of 127 mm. To ensure minimal resource use and time saving experimentation, Taguchi's L27 orthogonal array was used to design experimental matrix. During experimentation four input variables, i.e. cutting parameters were systematically varied at three levels as given in Table 2. Levels of variation for input variables were selected based on the manufacturer's recommendation and past experimentation considering that full cut should be achieved. More details regarding experimental and measurement procedures, details and results can be found in [7]. The experiment was performed to provide a solid knowledge-base about the influence of the laser cutting parameters on the cut quality characteristics (kerf width, surface roughness and width of HAZ) and material removal rate (MRR). The measurements of cut edge surface in terms of average surface roughness (Ra) were carried out with Surftest SJ-301 profilometer. Kerf width (w) and width of HAZ values were obtained using optical microscope Leitz. All measurements were repeated and recorded upon which averaged values were estimated. MRR is one of the most important criteria for determining the laser cutting operation, with a higher rate always preferred. This laser cutting performance characteristic was calculated as the product of cutting speed, workpiece thickness and kerf width. In the conducted experimental research it was observed that surface roughness ranges from 1.47 to 3.02 µm, kerf width from 0.29 to 0.55 mm, width of HAZ from 15 to 33.33 µm and MRR from 2118 to 4608 mm3/min. However, there is no experimental trial in which minimal surface roughness, minimal kerf width, minimal width of HAZ and maximal MRR is obtained at the same time. Combinations of laser cutting parameter values which are optimal for each criterion are summarized in Table 3. In order to obtain ranking of different experimental trials considering the afore-mentioned criteria, the PSI method was applied. Based on the complete ranking one can select the most suitable laser cutting process conditions for particular application.
⎡ x11 x21 X = [xij ]m × n = ⎢⎢… ⎣ xm1
Goal
v [m/min]
p [MPa]
min min min max
2.5 3 3 3
1.05 1.2 1.2 1.05
x1n ⎤ x2n ⎥ … ⎥ xmn ⎦
(1)
Step 3: Development of the normalized decision matrix in which the elements of the matrix are calculated using the following equations:
•
for maximization criteria:
xij =
•
xij xijmax
, i = 1, … m (2a)
for minimization criteria:
xij =
xijmin xij
, i = 1, … m
(2b)
Step 4: Determination of the mean values of normalized performances in relation to each criterion using the following equation: m
Ν=
1 ⋅ ∑ xij n i =1
(3)
Step 5: Determination of the values of the variation of preferences in relation to each criterion using the following equation: m
ϕj =
∑ (xij − Ν )2 i =1
(4)
Step 6: Determination of the deviations of the value of the preference in relation to each criterion using the following equation:
Ωj = 1 − ϕj
(5)
Step 7: Determination of criteria weights using the following equation: P [kW]
f [mm]
wj = Surface roughness Kerf width Width of HAZ MRR
… … … …
where xij is the performance value (attribute) of i-th alternative with respect to j-th criterion, and m and n are the number of alternatives and criteria, respectively.
Table 3 Best cutting conditions for each performance characteristic. Criteria
x12 x22 … xm2
2 1.8 2 1.6
−1.5 −0.5 −2.5 −2.5
Ωj n
∑ j =1 Ωj
(6)
Step 8: Determination of preference selection index values of alternatives using the following equation:
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θi =
∑ xij⋅wj (7)
j =1
Step 9: Based on the preference selection index values one needs to determine complete ranking of alternatives. The alternative which has the largest preference selection index value represents the best ranked alternative.
5. Results and discussion In the application of the PSI method it has to be noted that cut quality characteristics are minimization criteria, i.e. the goal is to obtain minimal performance values, while MRR is maximization criteria whereas higher values are preferable. The computational procedure of the PSI method is as follows. Decision matrix in this case is defined by 27 rows of experimental trials with different combinations of laser cutting parameter values (as per Taguchi's L27 orthogonal array) which can be regarded as alternatives and 4 columns of experimental results for average surface roughness (Ra), kerf width (w), width HAZ and MRR which can be regarded as criteria for assessment of alternatives. Initially, Eq. (2) was applied in order to obtain dimensionless values of alternative attribute values (performances) so that all these alternatives can be easily assessed. Then by applying Eq. (3), mean values of normalized performances in relation to each criterion were obtained as N=[0.75, 0.71, 0.73, 0.67]. Then by applying Eq. (4) values of the variation of preferences in relation to each criterion were calculated. Subsequently, by applying Eq. (5) deviations of the value of the preference in relation to each criterion were obtained as Ωj =[0.6919, 0.5634, 0.6116, 0.5118]. Thus, by applying Eq. (6) criteria weights were determined as w=[0.291, 0.237, 0.257, 0.215]. It can be observed that, surface roughness obtained the highest priority. It has to be noted that this method determines criteria weights based on objective approach, that is, only by using data given in decision matrix without incorporation of decision maker's preferences. Finally, by applying Eq. (7) preference selection index values of alternatives (combinations of laser cutting parameter values) were determined upon which complete ranking of alternatives was obtained. Table 4 gives the best three ranked alternatives and the two least preferred along with laser cutting conditions. From Table 2 it can be observed that experimental trial 27 can be regarded as the best alternative, i.e. it represents the best laser cutting condition. Also, one may observe that there is a very small difference in preference selection index values between trials 27 and 25, which are the first and the second ranked alternatives. It is observed that experimental trial 20 represents the least favored cutting condition since it has the smallest value of preference selection index. The laser cut surface pattern obtained under the best condition (experimental trial 27) is shown in Fig. 1. It is observed that focusing the laser beam deep into the bulk of material while using high assist gas pressure efficiently ejects the molten material from the kerf so the laser cutting is achieved without dross formation.
Fig. 1. Laser cut surface pattern obtained in experimental trial 27.
It is understood that relationships between preference selection index values and alternative attribute values with regard to the selected criteria is derived by the PSI method using Eqs. 2–7. However, in order to perform deeper analysis, development of mathematical relationship between alternative attribute values with regard to the selected criteria and preference selection index values was proposed. To this aim the mathematical model in the form of non-linear regression was developed. Thus by using experimental data from decision matrix and calculated preference selection index values the following mathematical model was developed:
θ = 1.65714 − 0.28412⋅Ra − 1.19108⋅w − 0.02523⋅HAZ + 0.00005 ⋅MRR − 0.04253⋅Ra2 + 0.91918⋅w 2 + 0.00038⋅HAZ 2
(8)
As could be expected the ANOVA analysis confirmed the validity of developed regression equation with p value of 0.000 and coefficient of multiple determination of R2=0.99. As a consequence of this equation it is observed that complete ranking of alternatives by the PSI method is based on linear and quadratic transformations of data contained in the decision matrix without considering any possible interaction effects. The derived mathematical model for prediction of the preference selection index value suggests that there is no significant interactions between criteria. In other words, the effect of alternatives attribute values of a given criteria on the preference selection index value is not dependent, in qualitative sense, on the alternative attribute values to another criterion. This is evident considering all possible interaction graphs given in Fig. 2. Once the mathematical model is developed one can predict preference selection index values for any combination of alternative attribute values within the experimental hyperspace. Thus, maximization of the derived equation would give the ideal solution while minimization gives the anti-ideal solution. It is clear that maximal preference selection index value is obtained when kerf width, surface roughness and width of HAZ values are minimal, while MRR values are maximal. Substituting appropriate values of surface roughness, kerf width, HAZ and MRR into Eq. (8) one obtainsθ = 1. On the other hand, for anti-ideal solution one obtainsθ = 0.5.
Table 4 Ranking of experimental trials. Laser cutting conditions
Criteria
Experimental trial as per Taguchi's L27 orthogonal array
v [m/min]
p [MPa]
P [kW]
f [mm]
Ra [µm]
w [mm]
Width of HAZ [µm]
MRR [mm3/min]
θi
Ranking
27 25 18 – – 19 20
3 3 3 – – 2 2
1.2 0.9 1.2 – – 0.9 1.05
2 2 1.8 – – 2 2
−2,50 −1,50 −0.5 – – −0.5 −2.5
1.93 1.6 1.91 – – 1.89 3.02
0.44 0.39 0.29 – – 0.38 0.54
15 18.33 19.33 – – 28.33 19.33
3987 3501 2583 – – 2256 3252
0.818 0.815 0.78 – – 0.648 0.618
1 2 3 – – 26 27
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Fig. 2. Interaction effects of criteria on preference selection index.
consequence of Eq. (8). From Fig. 3 the following can be observed regarding selection of the best alternatives:
Quantitative effects of alternative attribute values for each criterion on preference selection index values are given in Fig. 3. The plots are obtained by changing one alternative performance attribute values with respect to one criterion at a time, while keeping all other alternative attribute values with respect to other criteria constant at low, center and high level. Linear effect of alternative attribute values for MRR on preference selection index values (Fig. 3d) is evident and is a logical
• 218
For any alternative attribute values with respect to Ra, it is prefferable to chose combination of maximal alternative attribute values with respect to MRR, w and HAZ, but
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Fig. 3. Effects of criteria on preference selection index (― - alternative attribute values with respect to other criteria constant at center level, - alternative attribute values with respect to other criteria constant at high level). respect to other criteria constant at low level,
•
developed regression equation with p value of 0.000 and coefficient of multiple determination of R2=0.82. The mean absolute percentage error between estimated and predicted values of preference selection index is found to be 2.5% which further confirms the validity of the developed mathematical model. Unlike the mathematical model given in Eq. (8) it has been observed that there exists a qualitative changes in preference selection index values when values of laser cutting parameters are changed. Particularly, the effect of laser power and focal position must be considered through interaction with assist gas pressure. By optimizing Eq. (9) it was observed that maximal performance selection index value of θ =0.8183 corresponds to experimental trial 27. In other words, the determined solution by continual and discrete optimization (by the PSI method) is the same. For the analysis of preference selection index values obtained in experimental research histogram was ploted. The distribution of preference selection index values in the form of histogram in eleven equal ranges, between minimum and maximum values (0.5 and 1), is given in Fig. 4. It can be observed from Fig. 4 that preference selection index values have a mean value and standard deviation of 0.7194 and 0.056, respectively. The results show that about 40% of the experimental trials have preference selection index values that are greater than the average preference selection index value of 0.75. Regarding the application of the PSI method one needs to
For any alternative attribute values with respect to w, HAZ or MRR, it is prefferable to chose combination of minimal alternative attribute values with respect to Ra, w, HAZ and MRR.
The previously developed mathematical model established relationship between preference selection index values and alternative attribute values with regard to the selected criteria and it was useful regarding the analysis of the correlations between alternative performance attribute values with respect to selected criteria. However, for practical application of the PSI method in real manufacturing environment it is of utmost importance to model the relationships between performance selection index values and laser cutting parameter values. In such way it would be possible to search for the most suitable combination of laser cutting parameter values so as to maximize performance selection index value. This is particularly useful in planning and control of laser cutting operation. Therefore, by using the calculated preference selection index values the following mathematical model was developed:
θ = 1.193 − 0.277⋅P + 0.116⋅v − 0.898⋅p + 0.032⋅f − 0.09⋅P2 − 0.055 ⋅v 2 + 0.201⋅p 2 − 0.017⋅f 2 + 0.156⋅P⋅v + 0.24⋅P⋅p + 0.033⋅P⋅f − 0.042 ⋅v⋅p − 0.023⋅v⋅f − 0.097⋅p⋅f
- alternative attribute values with
(9)
As could be expected the ANOVA analysis confirmed the validity of 219
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calculation of lower and upper preference selection index values is enabled. The discussed methodology is found to be very useful in real manufacturing environment since it involves simple calculations which are easy to understand and implement. However, while applying the PSI method it was observed that it cannot be effective in situations where there exist a large number of alternatives which have attribute values (performances) very close to preferred. References [1] B.S. Yilbas, Effect of process parameters on the kerf width during the laser cutting process, Proc. Inst. Mech. Eng. Part B: J. Eng. Manuf. 215 (10) (2001) 1357–1365. [2] M.J. Tsai, C.H. Li, C.C. Chen, Optimal laser-cutting parameters for QFN packages by utilizing artificial neural networks and genetic algorithm, J. Mater. Process. Technol. 208 (1–3) (2008) 270–283. [3] C.B. Yang, D.S. Deng, H.L. Chiang, Combining the Taguchi method with artificial neural network to construct a prediction model of a CO2 laser cutting experiment, Int. J. Adv. Manuf. Technol. 59 (9) (2011) 1103–1111. [4] A.K. Dubey, V. Yadava, Multi-objective optimisation of laser beam cutting process, Opt. Laser Technol. 40 (3) (2008) 562–570. [5] U. Caydaş, A. Hasçalik, Use of the grey relational analysis to determine optimum laser cutting parameters with multi-performance characteristics, Opt. Laser Technol. 40 (7) (2008) 987–994. [6] A. Sharma, V. Yadava, Modelling and optimization of cut quality during pulsed Nd: yag laser cutting of thin Al-alloy sheet for straight profile, Opt. Laser Technol. 44 (1) (2012) 159–168. [7] M. Madić, M. Radovanović, M. Manić, M. Trajanović, Optimization of ANN models using different optimization methods for improving CO2 laser cut quality characteristics, J. Braz. Soc. Mech. Sci. Eng. 36 (1) (2014) 91–99. [8] B.S. Yilbas, S.J. Hyder, M. Sunar, The taguchi method for determining CO2 laser cut quality, J. Laser Appl. 10 (2) (1998) 71–77. [9] T.A. El-Taweel, A.M. Abdel-Maaboud, B.S. Azzam, A.E. Mohammad, Parametric studies on the CO2 laser cutting of kevlar-49 composite, Int. J. Adv. Manuf. Technol. 40 (9–10) (2009) 907–917. [10] A. Hasçalık, M. Ay, CO2 laser cut quality of Inconel 718 nickel - based superalloy, Opt. Laser Technol. 48 (1) (2013) 554–564. [11] Z. Durukan, A.R. Motorcu, A. Güllü, Modeling of the effects of parameters on dimensional accuracy in laser cutting of AISI 304 steel with different geometries, J. Fac. Eng. Archit. Gazi Univ. 29 (3) (2014) 505–515. [12] G. Norkey, A.K. Dubey, S. Agrawal, Artificial intelligence based modeling and optimization of heat affected zone in Nd: yag laser cutting of duralumin sheet, J. Intell. Fuzzy Syst. 27 (3) (2014) 1545–1555. [13] L.D. Scintilla, Continuous-wave fiber laser cutting of aluminum thin sheets: effect of process parameters and optimization, Opt. Eng. 53 (6) (2014) (066113-066113). [14] Y. Nukman, M.A. Hassan, M.Z. Harizam, Optimization of prediction error in CO2 laser cutting process by taguchi artificial neural network hybrid with genetic algorithm, Appl. Math. Inf. Sci. 7 (1) (2013) 363–370. [15] D. Kondayya, A. Gopala Krishna, An integrated evolutionary approach for modelling and optimization of laser beam cutting process, Int. J. Adv. Manuf. Technol. 65 (1–4) (2013) 259–274. [16] H.J. Hao, J.Y. Xu, J. Li, Prediction of laser cutting quality based on an improved Pareto genetic algorithm, Lasers Eng. 27 (1–2) (2014) 43–56. [17] E.K. Zavadskas, T. Vilutienė, Z. Turskis, J. Šaparauskas, Multi-criteria analysis of projects' performance in construction, Arch. Civ. Mech. Eng. 14 (1) (2014) 114–121. [18] K. Maniya, M.G. Bhatt, A selection of material using a novel type decision-making method: preference selection index method, Mater. Des. 31 (4) (2010) 1785–1789. [19] B. Vahdani, S.M. Mousavi, S. Ebrahimnejad, Soft computing-based preference selection index method for human resource management, J. Intell. Fuzzy Syst. 26 (1) (2014) 393–403. [20] R. Attri, S. Grover, Application of preference selection index method for decision making over the design stage of production system life cycle, J. King Saud Univ.Eng. Sci. 27 (2) (2015) 207–216.
Fig. 4. Distribution of preference selection index values obtained in experimental research.
emphasize the following. Although some authors have noted that the PSI method can be applied successfully to any number of alternatives [18,20], the application of the PSI method is not convenient in some cases. Namely, initially the assessment of laser cuts was planned to be performed by considering also the taper angle and burr height. However, it has been observed that in situations where a number of alternatives have attribute values which are very close to those which are preferred, deviations of the value of the preference, as determined by Eq. (5), are negative numbers and subsequent application of following formulae for determination of criteria weights is not possible. 6. Conclusion This paper discussed four main approaches for determination of process conditions for improvement of multiple quality and productivity characteristics at the same time in laser cutting. Particular emphasize has been given to proposed algorithmic procedure, which is based on the application the PSI method, for determination of the most suitable laser cutting process conditions. As the MCDM method which provides objective approach for criteria weights determination, ranking of alternatives considered in decision matrix is solely based on the data contained in the decision matrix. More precisely, complete ranking of alternatives by the PSI method is based on linear and quadratic transformations of data contained in the decision matrix. Although the PSI method, in essence, belongs to a class of MCDM methods which provide discrete optimization solutions, in this paper an approach based on regression analysis for model development was proposed in order to: (i) model the preference selection index values considering alternative attribute values with regard to the selected criteria in order to analyze the correlations between criteria, (ii) model the preference selection index considering laser cutting parameter values with an aim to predict preference selection index values for any combination of laser cutting parameter values. Regression based modeling of preference selection index values has advantages because
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Full length article
Determination of laser cutting process conditions using the preference selection index method
crossmark
⁎
Miloš Madića, , Jurgita Antuchevicieneb, Miroslav Radovanovića, Dušan Petkovića a b
Faculty of Mechanical Engineering, University of Niš, A. Medvedeva 14, 18000 Niš, Serbia Vilnius Gediminas Technical University, Sauletekio av. 11, LT-10223 Vilnius, Lithuania
A R T I C L E I N F O
A BS T RAC T
Keywords: Laser cutting Optimization MCDM Cut quality Preference selection index method
Determination of adequate parameter settings for improvement of multiple quality and productivity characteristics at the same time is of great practical importance in laser cutting. This paper discusses the application of the preference selection index (PSI) method for discrete optimization of the CO2 laser cutting of stainless steel. The main motivation for application of the PSI method is that it represents an almost unexplored multi-criteria decision making (MCDM) method, and moreover, this method does not require assessment of the considered criteria relative significances. After reviewing and comparing the existing approaches for determination of laser cutting parameter settings, the application of the PSI method was explained in detail. Experiment realization was conducted by using Taguchi's L27 orthogonal array. Roughness of the cut surface, heat affected zone (HAZ), kerf width and material removal rate (MRR) were considered as optimization criteria. The proposed methodology is found to be very useful in real manufacturing environment since it involves simple calculations which are easy to understand and implement. However, while applying the PSI method it was observed that it can not be useful in situations where there exist a large number of alternatives which have attribute values (performances) very close to those which are preferred.
1. Introduction Laser cutting is one of the industry leading technologies for cutting a wide variety of materials. Compared to other alternative cutting technologies, laser cutting offers significant advantages and possibilities such as ability to cut complex geometries with tight tolerances, high cutting speeds, i.e. increased productivity, localized heat affected zone (HAZ), high quality, ease of automation, etc. [1–3]. Although initial capital investments of laser cutting technology are high, operational costs are low making this technology being economically competitive and cost effective. In order to take full advantages and benefits that laser cutting technology offers one needs to carefully consider determination and selection of adequate process conditions, i.e. particular combination of the main parameter values. In general, the selection of parameter settings is mainly dependent on the composition of the workpiece material, workpiece thickness and desired performance characteristics related to cost, quality and productivity. However, the main difficulty is the fact that optimal combination of laser cutting parameter values for one performance characteristic is not even near optimal for other performance characteristics [4–7]. Therefore, determination of laser ⁎
cutting process conditions for multi-performance (multi-criteria) optimization is of prime importance. In the open literature and manufacturing practice one can identify four main approaches for determining of the most suitable laser cutting conditions for a given application: trial and error method, Taguchi method, continual optimization and discrete optimization. Trial and error method is one the most common approaches in real manufacturing environment. It assumes that process planners and engineers use acquired experience and recommendations from handbooks for selecting laser cutting parameter values. This subjective approach leads to the fact that selected values of laser cutting parameters vary from one case to another and are preferably conservative. Although this approach may be sufficient in most applications, it denies the possibility of using better (optimized) cutting conditions and ultimately it does not provide a basis for the full utilization of the laser cutting machine. It can be argued that, due to complexity and stochastic nature of laser cutting, even highly skilled process planners and engineers can hardly determine laser cutting process conditions with respect to several criteria. Due to its simplicity and ease of implementation the application of the Taguchi method is the second most common approach in real
Corresponding author. E-mail addresses: [email protected] (M. Madić), [email protected] (J. Antucheviciene), [email protected] (M. Radovanović), [email protected] (D. Petković). http://dx.doi.org/10.1016/j.optlastec.2016.10.005 Received 24 August 2016; Accepted 24 October 2016 0030-3992/ © 2016 Elsevier Ltd. All rights reserved.
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manufacturing environment. The Taguchi method is a well-known, unique and powerful method for product/process quality improvement. In laser cutting, application of this method allows for identification of near optimal laser cutting process conditions making the process insensitive to noise factors such as environmental temperature, humidity or dust. High popularity of this method is due to fact that this approach does not require development of any mathematical model, thus can be readily applied in real manufacturing environment [4– 6,8,9]. However, on the other hand, this approach permits only discrete optimization and if one need to consider multiple objectives, application of additional methods such as utility method, principal component analysis or principal component analysis is inevitable. Determination of laser cutting process conditions based on continual optimization represents higher level approach. It consists of development of an empirical model of a laser cutting performance characteristic and application of an optimization method. Empirical mathematical models establish relationships between inputs (cutting parameters) and outputs (performance characteristics). To this aim regression analysis, artificial neural networks and genetic programming are predominantly being used [3,7,10–13]. Once developed, these mathematical models are used as objective functions which are optimized using an optimization method. Since laser cutting process optimization problems are complex, highly non-linear and multidimensional, meta-heuristic optimization methods have become a preferred trend for solving these types of optimization problems [2,7,12,14–16]. Integration of empirical models and optimization methods allows for continuous single and multi-criteria optimization of laser cutting process. Namely, laser cutting parameter values (independent variables) are defined in continual or integer domain and the goal is to determine the best solution (laser cutting process condition) which satisfies all previously set constraints and ranges of independent variables. Although providing better optimization solutions this approach is more time and computationally expensive and requires higher knowledge levels from design of experiments, mathematical model development, optimization theory as well as metaheuristics. Finally, the problem of determining of laser cutting process conditions can be viewed as a multi-criteria decision making (MCDM) problem in which a particular cutting conditions represents an alternative while performance characteristics represent criteria upon which alternatives are assessed [17]. Although there is a number of mathematically relatively simple MCDM methods for assessment and ranking of alternatives, determination of laser cutting process conditions by using MCDM methods still do not have wide application in practice. Given that there is a finite number of pre-known alternatives (laser cutting process conditions) this type of multi-criteria optimization problems are referred as discrete. The previously discussed four main approaches are predominantly used in manufacturing practice for determination of laser cutting process conditions. In practice the selection of a given approach is dependent on the particular application. Thus, if there is a need for large batch laser cutting on expensive work materials, the application of continual optimization may the right choice. On the other hand, in situations where waste and possible post-processing imply negligible financial losses, one can select other more simple approaches. Each approach has some advantages, disadvantages and limitations. Although does not provide even near optimal laser cutting process conditions, trial and error method can provide satisfactory results without the need of domain expert knowledge. On the other hand, formulation and solving of multi-criteria continual optimization problems can provide optimal laser cutting process conditions, however, this approach requires a considerable knowledge level of laser cutting, DOE, mathematical modeling, optimization and AI methods. Finally, if time is a limiting factor one can consider discrete optimization approach, whether Taguchi or MCDM methods. Comparison of existing approaches for determination of laser cutting process conditions with
Table 1 Comparison of the existing approaches for determination of laser cutting process conditions.
Computationally expense Implementation time Quality of determined solution Practical suitability Requirements of particular knowledge Possibility to handle a number of criteria Requirements of using specialized software
Trial and error
Taguchi method
Continual optimization
Discrete optimization
–
small
high
small
permanent
fast
average
fast
low
good
high
good
high –
very high some
very high high
average some
none
good
good
very good
–
none
yes
none
respect to different criteria is given in Table 1.
2. Application methodology The application methodology for determination of laser cutting process conditions with the use of the PSI method can be summarized in the six following steps: problem definition, pre-analysis, laser cutting experimental investigation, data acquisition, formulation of decision matrix and application of the PSI method. Problem definition implies determination of the objectives (customer requirements and manufacturers preferences) regarding desired cut quality, productivity and cost. The goal of pre-analysis step is identification of main laser cutting parameters that predominantly influence the objective functions, i.e. laser cutting performance characteristics (surface roughness, kerf width, HAZ, MRR, etc.). Therefore, the laser cutting performance characteristics can be regarded as criteria for evaluation of laser cuts. To this aim in this stage one can perform one factor at a time (OFAT) experimental trials or conduct screening experimental designs such as Plackett-Burman designs. Once the main laser cutting parameters are singled out, experimental trials with different laser cutting parameter values combinations are to be carried out. The laser cutting parameters are to be varied in a range considering manufacturing practices so that wider experimental hyperspace is covered while ensuring that full laser cut is obtained in each combination. To ensure minimal resource use and time saving experimentation one could apply Taguchi's orthogonal arrays, Box-Behnken designs, composite designs, fractional factorial designs, etc. Data acquisition refers to measurement/calculation of quality and performance characteristics that are obtained in each experimental trial, i.e. for each laser cutting parameter values combination. After collection of experimental data one needs to formulate decision matrix based on experimental data for each experimental trial, i.e. combination of laser cutting parameter values with respect to all considered criteria (performance characteristics). Each row in the decision matrix represents one alternative (specific laser cutting process condition), and each column represents one criterion. The final step is the application of the PSI method for deriving the decision rule upon which the most suitable laser cutting process condition can be determined. The goal of the application of any MCDM method is to determine aggregate function, so-called decision 215
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4. PSI method
Table 2 Levels of variation for input variables. Input variable
Level 1
Level 2
Level 3
Cutting speed, v [m/min] Assist gas pressure, p [MPa] Laser power, P [kW] Focal position, f [mm]
2 0.9 1.6 −2.5
2.5 1.05 1.8 −1.5
3 1.2 2 −0.5
The PSI method was proposed by Maniya and Bhatt [18] in 2010 for solving material selection MCDM problems. In 2014, Vahdani et al. [19] proposed interval valued fuzzy PSI method in order to solve complex decision problems under uncertainty. Unlike the most MCDM methods, the PSI method does not require determination of the relative importance of the criteria, and therefore it is not necessary to determine criteria weights. Thus, the method is particularly useful in cases where a conflict in deciding the relative importance among criteria appears [20]. Actually, the PSI method determines criteria weights only by using information provided in the decision matrix, i.e. it uses objective approach to determine criteria weights like standard deviation or entropy method. The application methodology of the PSI method for solving MCDM problems includes several steps [18–20]:
rule, which shows the overall assessment of alternatives using data from the decision matrix as well as considering the preferences of the decision maker regarding relative significances of considered criteria. Based on aggregate function values of alternatives it is possible to obtain complete ranking of alternatives. 3. Experimental details
Step 1: Identification of the relevant criteria for evaluation of the alternatives. Step 2: Development of the initial decision matrix, X:
The CO2 laser cutting experiment was carried out using ByVention 3015 CO2 laser cutting machine. The workpiece material was AISI 304 stainless steel and the sheet thickness was 3 mm. Nitrogen with purity of 99.95% was used as assist gas in experimentation and it was supplied via conical nozzle with 2 mm diameter. The laser beam was focused using lens with focal length of 127 mm. To ensure minimal resource use and time saving experimentation, Taguchi's L27 orthogonal array was used to design experimental matrix. During experimentation four input variables, i.e. cutting parameters were systematically varied at three levels as given in Table 2. Levels of variation for input variables were selected based on the manufacturer's recommendation and past experimentation considering that full cut should be achieved. More details regarding experimental and measurement procedures, details and results can be found in [7]. The experiment was performed to provide a solid knowledge-base about the influence of the laser cutting parameters on the cut quality characteristics (kerf width, surface roughness and width of HAZ) and material removal rate (MRR). The measurements of cut edge surface in terms of average surface roughness (Ra) were carried out with Surftest SJ-301 profilometer. Kerf width (w) and width of HAZ values were obtained using optical microscope Leitz. All measurements were repeated and recorded upon which averaged values were estimated. MRR is one of the most important criteria for determining the laser cutting operation, with a higher rate always preferred. This laser cutting performance characteristic was calculated as the product of cutting speed, workpiece thickness and kerf width. In the conducted experimental research it was observed that surface roughness ranges from 1.47 to 3.02 µm, kerf width from 0.29 to 0.55 mm, width of HAZ from 15 to 33.33 µm and MRR from 2118 to 4608 mm3/min. However, there is no experimental trial in which minimal surface roughness, minimal kerf width, minimal width of HAZ and maximal MRR is obtained at the same time. Combinations of laser cutting parameter values which are optimal for each criterion are summarized in Table 3. In order to obtain ranking of different experimental trials considering the afore-mentioned criteria, the PSI method was applied. Based on the complete ranking one can select the most suitable laser cutting process conditions for particular application.
⎡ x11 x21 X = [xij ]m × n = ⎢⎢… ⎣ xm1
Goal
v [m/min]
p [MPa]
min min min max
2.5 3 3 3
1.05 1.2 1.2 1.05
x1n ⎤ x2n ⎥ … ⎥ xmn ⎦
(1)
Step 3: Development of the normalized decision matrix in which the elements of the matrix are calculated using the following equations:
•
for maximization criteria:
xij =
•
xij xijmax
, i = 1, … m (2a)
for minimization criteria:
xij =
xijmin xij
, i = 1, … m
(2b)
Step 4: Determination of the mean values of normalized performances in relation to each criterion using the following equation: m
Ν=
1 ⋅ ∑ xij n i =1
(3)
Step 5: Determination of the values of the variation of preferences in relation to each criterion using the following equation: m
ϕj =
∑ (xij − Ν )2 i =1
(4)
Step 6: Determination of the deviations of the value of the preference in relation to each criterion using the following equation:
Ωj = 1 − ϕj
(5)
Step 7: Determination of criteria weights using the following equation: P [kW]
f [mm]
wj = Surface roughness Kerf width Width of HAZ MRR
… … … …
where xij is the performance value (attribute) of i-th alternative with respect to j-th criterion, and m and n are the number of alternatives and criteria, respectively.
Table 3 Best cutting conditions for each performance characteristic. Criteria
x12 x22 … xm2
2 1.8 2 1.6
−1.5 −0.5 −2.5 −2.5
Ωj n
∑ j =1 Ωj
(6)
Step 8: Determination of preference selection index values of alternatives using the following equation:
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θi =
∑ xij⋅wj (7)
j =1
Step 9: Based on the preference selection index values one needs to determine complete ranking of alternatives. The alternative which has the largest preference selection index value represents the best ranked alternative.
5. Results and discussion In the application of the PSI method it has to be noted that cut quality characteristics are minimization criteria, i.e. the goal is to obtain minimal performance values, while MRR is maximization criteria whereas higher values are preferable. The computational procedure of the PSI method is as follows. Decision matrix in this case is defined by 27 rows of experimental trials with different combinations of laser cutting parameter values (as per Taguchi's L27 orthogonal array) which can be regarded as alternatives and 4 columns of experimental results for average surface roughness (Ra), kerf width (w), width HAZ and MRR which can be regarded as criteria for assessment of alternatives. Initially, Eq. (2) was applied in order to obtain dimensionless values of alternative attribute values (performances) so that all these alternatives can be easily assessed. Then by applying Eq. (3), mean values of normalized performances in relation to each criterion were obtained as N=[0.75, 0.71, 0.73, 0.67]. Then by applying Eq. (4) values of the variation of preferences in relation to each criterion were calculated. Subsequently, by applying Eq. (5) deviations of the value of the preference in relation to each criterion were obtained as Ωj =[0.6919, 0.5634, 0.6116, 0.5118]. Thus, by applying Eq. (6) criteria weights were determined as w=[0.291, 0.237, 0.257, 0.215]. It can be observed that, surface roughness obtained the highest priority. It has to be noted that this method determines criteria weights based on objective approach, that is, only by using data given in decision matrix without incorporation of decision maker's preferences. Finally, by applying Eq. (7) preference selection index values of alternatives (combinations of laser cutting parameter values) were determined upon which complete ranking of alternatives was obtained. Table 4 gives the best three ranked alternatives and the two least preferred along with laser cutting conditions. From Table 2 it can be observed that experimental trial 27 can be regarded as the best alternative, i.e. it represents the best laser cutting condition. Also, one may observe that there is a very small difference in preference selection index values between trials 27 and 25, which are the first and the second ranked alternatives. It is observed that experimental trial 20 represents the least favored cutting condition since it has the smallest value of preference selection index. The laser cut surface pattern obtained under the best condition (experimental trial 27) is shown in Fig. 1. It is observed that focusing the laser beam deep into the bulk of material while using high assist gas pressure efficiently ejects the molten material from the kerf so the laser cutting is achieved without dross formation.
Fig. 1. Laser cut surface pattern obtained in experimental trial 27.
It is understood that relationships between preference selection index values and alternative attribute values with regard to the selected criteria is derived by the PSI method using Eqs. 2–7. However, in order to perform deeper analysis, development of mathematical relationship between alternative attribute values with regard to the selected criteria and preference selection index values was proposed. To this aim the mathematical model in the form of non-linear regression was developed. Thus by using experimental data from decision matrix and calculated preference selection index values the following mathematical model was developed:
θ = 1.65714 − 0.28412⋅Ra − 1.19108⋅w − 0.02523⋅HAZ + 0.00005 ⋅MRR − 0.04253⋅Ra2 + 0.91918⋅w 2 + 0.00038⋅HAZ 2
(8)
As could be expected the ANOVA analysis confirmed the validity of developed regression equation with p value of 0.000 and coefficient of multiple determination of R2=0.99. As a consequence of this equation it is observed that complete ranking of alternatives by the PSI method is based on linear and quadratic transformations of data contained in the decision matrix without considering any possible interaction effects. The derived mathematical model for prediction of the preference selection index value suggests that there is no significant interactions between criteria. In other words, the effect of alternatives attribute values of a given criteria on the preference selection index value is not dependent, in qualitative sense, on the alternative attribute values to another criterion. This is evident considering all possible interaction graphs given in Fig. 2. Once the mathematical model is developed one can predict preference selection index values for any combination of alternative attribute values within the experimental hyperspace. Thus, maximization of the derived equation would give the ideal solution while minimization gives the anti-ideal solution. It is clear that maximal preference selection index value is obtained when kerf width, surface roughness and width of HAZ values are minimal, while MRR values are maximal. Substituting appropriate values of surface roughness, kerf width, HAZ and MRR into Eq. (8) one obtainsθ = 1. On the other hand, for anti-ideal solution one obtainsθ = 0.5.
Table 4 Ranking of experimental trials. Laser cutting conditions
Criteria
Experimental trial as per Taguchi's L27 orthogonal array
v [m/min]
p [MPa]
P [kW]
f [mm]
Ra [µm]
w [mm]
Width of HAZ [µm]
MRR [mm3/min]
θi
Ranking
27 25 18 – – 19 20
3 3 3 – – 2 2
1.2 0.9 1.2 – – 0.9 1.05
2 2 1.8 – – 2 2
−2,50 −1,50 −0.5 – – −0.5 −2.5
1.93 1.6 1.91 – – 1.89 3.02
0.44 0.39 0.29 – – 0.38 0.54
15 18.33 19.33 – – 28.33 19.33
3987 3501 2583 – – 2256 3252
0.818 0.815 0.78 – – 0.648 0.618
1 2 3 – – 26 27
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Fig. 2. Interaction effects of criteria on preference selection index.
consequence of Eq. (8). From Fig. 3 the following can be observed regarding selection of the best alternatives:
Quantitative effects of alternative attribute values for each criterion on preference selection index values are given in Fig. 3. The plots are obtained by changing one alternative performance attribute values with respect to one criterion at a time, while keeping all other alternative attribute values with respect to other criteria constant at low, center and high level. Linear effect of alternative attribute values for MRR on preference selection index values (Fig. 3d) is evident and is a logical
• 218
For any alternative attribute values with respect to Ra, it is prefferable to chose combination of maximal alternative attribute values with respect to MRR, w and HAZ, but
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Fig. 3. Effects of criteria on preference selection index (― - alternative attribute values with respect to other criteria constant at center level, - alternative attribute values with respect to other criteria constant at high level). respect to other criteria constant at low level,
•
developed regression equation with p value of 0.000 and coefficient of multiple determination of R2=0.82. The mean absolute percentage error between estimated and predicted values of preference selection index is found to be 2.5% which further confirms the validity of the developed mathematical model. Unlike the mathematical model given in Eq. (8) it has been observed that there exists a qualitative changes in preference selection index values when values of laser cutting parameters are changed. Particularly, the effect of laser power and focal position must be considered through interaction with assist gas pressure. By optimizing Eq. (9) it was observed that maximal performance selection index value of θ =0.8183 corresponds to experimental trial 27. In other words, the determined solution by continual and discrete optimization (by the PSI method) is the same. For the analysis of preference selection index values obtained in experimental research histogram was ploted. The distribution of preference selection index values in the form of histogram in eleven equal ranges, between minimum and maximum values (0.5 and 1), is given in Fig. 4. It can be observed from Fig. 4 that preference selection index values have a mean value and standard deviation of 0.7194 and 0.056, respectively. The results show that about 40% of the experimental trials have preference selection index values that are greater than the average preference selection index value of 0.75. Regarding the application of the PSI method one needs to
For any alternative attribute values with respect to w, HAZ or MRR, it is prefferable to chose combination of minimal alternative attribute values with respect to Ra, w, HAZ and MRR.
The previously developed mathematical model established relationship between preference selection index values and alternative attribute values with regard to the selected criteria and it was useful regarding the analysis of the correlations between alternative performance attribute values with respect to selected criteria. However, for practical application of the PSI method in real manufacturing environment it is of utmost importance to model the relationships between performance selection index values and laser cutting parameter values. In such way it would be possible to search for the most suitable combination of laser cutting parameter values so as to maximize performance selection index value. This is particularly useful in planning and control of laser cutting operation. Therefore, by using the calculated preference selection index values the following mathematical model was developed:
θ = 1.193 − 0.277⋅P + 0.116⋅v − 0.898⋅p + 0.032⋅f − 0.09⋅P2 − 0.055 ⋅v 2 + 0.201⋅p 2 − 0.017⋅f 2 + 0.156⋅P⋅v + 0.24⋅P⋅p + 0.033⋅P⋅f − 0.042 ⋅v⋅p − 0.023⋅v⋅f − 0.097⋅p⋅f
- alternative attribute values with
(9)
As could be expected the ANOVA analysis confirmed the validity of 219
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calculation of lower and upper preference selection index values is enabled. The discussed methodology is found to be very useful in real manufacturing environment since it involves simple calculations which are easy to understand and implement. However, while applying the PSI method it was observed that it cannot be effective in situations where there exist a large number of alternatives which have attribute values (performances) very close to preferred. References [1] B.S. Yilbas, Effect of process parameters on the kerf width during the laser cutting process, Proc. Inst. Mech. Eng. Part B: J. Eng. Manuf. 215 (10) (2001) 1357–1365. [2] M.J. Tsai, C.H. Li, C.C. Chen, Optimal laser-cutting parameters for QFN packages by utilizing artificial neural networks and genetic algorithm, J. Mater. Process. Technol. 208 (1–3) (2008) 270–283. [3] C.B. Yang, D.S. Deng, H.L. Chiang, Combining the Taguchi method with artificial neural network to construct a prediction model of a CO2 laser cutting experiment, Int. J. Adv. Manuf. Technol. 59 (9) (2011) 1103–1111. [4] A.K. Dubey, V. Yadava, Multi-objective optimisation of laser beam cutting process, Opt. Laser Technol. 40 (3) (2008) 562–570. [5] U. Caydaş, A. Hasçalik, Use of the grey relational analysis to determine optimum laser cutting parameters with multi-performance characteristics, Opt. Laser Technol. 40 (7) (2008) 987–994. [6] A. Sharma, V. Yadava, Modelling and optimization of cut quality during pulsed Nd: yag laser cutting of thin Al-alloy sheet for straight profile, Opt. Laser Technol. 44 (1) (2012) 159–168. [7] M. Madić, M. Radovanović, M. Manić, M. Trajanović, Optimization of ANN models using different optimization methods for improving CO2 laser cut quality characteristics, J. Braz. Soc. Mech. Sci. Eng. 36 (1) (2014) 91–99. [8] B.S. Yilbas, S.J. Hyder, M. Sunar, The taguchi method for determining CO2 laser cut quality, J. Laser Appl. 10 (2) (1998) 71–77. [9] T.A. El-Taweel, A.M. Abdel-Maaboud, B.S. Azzam, A.E. Mohammad, Parametric studies on the CO2 laser cutting of kevlar-49 composite, Int. J. Adv. Manuf. Technol. 40 (9–10) (2009) 907–917. [10] A. Hasçalık, M. Ay, CO2 laser cut quality of Inconel 718 nickel - based superalloy, Opt. Laser Technol. 48 (1) (2013) 554–564. [11] Z. Durukan, A.R. Motorcu, A. Güllü, Modeling of the effects of parameters on dimensional accuracy in laser cutting of AISI 304 steel with different geometries, J. Fac. Eng. Archit. Gazi Univ. 29 (3) (2014) 505–515. [12] G. Norkey, A.K. Dubey, S. Agrawal, Artificial intelligence based modeling and optimization of heat affected zone in Nd: yag laser cutting of duralumin sheet, J. Intell. Fuzzy Syst. 27 (3) (2014) 1545–1555. [13] L.D. Scintilla, Continuous-wave fiber laser cutting of aluminum thin sheets: effect of process parameters and optimization, Opt. Eng. 53 (6) (2014) (066113-066113). [14] Y. Nukman, M.A. Hassan, M.Z. Harizam, Optimization of prediction error in CO2 laser cutting process by taguchi artificial neural network hybrid with genetic algorithm, Appl. Math. Inf. Sci. 7 (1) (2013) 363–370. [15] D. Kondayya, A. Gopala Krishna, An integrated evolutionary approach for modelling and optimization of laser beam cutting process, Int. J. Adv. Manuf. Technol. 65 (1–4) (2013) 259–274. [16] H.J. Hao, J.Y. Xu, J. Li, Prediction of laser cutting quality based on an improved Pareto genetic algorithm, Lasers Eng. 27 (1–2) (2014) 43–56. [17] E.K. Zavadskas, T. Vilutienė, Z. Turskis, J. Šaparauskas, Multi-criteria analysis of projects' performance in construction, Arch. Civ. Mech. Eng. 14 (1) (2014) 114–121. [18] K. Maniya, M.G. Bhatt, A selection of material using a novel type decision-making method: preference selection index method, Mater. Des. 31 (4) (2010) 1785–1789. [19] B. Vahdani, S.M. Mousavi, S. Ebrahimnejad, Soft computing-based preference selection index method for human resource management, J. Intell. Fuzzy Syst. 26 (1) (2014) 393–403. [20] R. Attri, S. Grover, Application of preference selection index method for decision making over the design stage of production system life cycle, J. King Saud Univ.Eng. Sci. 27 (2) (2015) 207–216.
Fig. 4. Distribution of preference selection index values obtained in experimental research.
emphasize the following. Although some authors have noted that the PSI method can be applied successfully to any number of alternatives [18,20], the application of the PSI method is not convenient in some cases. Namely, initially the assessment of laser cuts was planned to be performed by considering also the taper angle and burr height. However, it has been observed that in situations where a number of alternatives have attribute values which are very close to those which are preferred, deviations of the value of the preference, as determined by Eq. (5), are negative numbers and subsequent application of following formulae for determination of criteria weights is not possible. 6. Conclusion This paper discussed four main approaches for determination of process conditions for improvement of multiple quality and productivity characteristics at the same time in laser cutting. Particular emphasize has been given to proposed algorithmic procedure, which is based on the application the PSI method, for determination of the most suitable laser cutting process conditions. As the MCDM method which provides objective approach for criteria weights determination, ranking of alternatives considered in decision matrix is solely based on the data contained in the decision matrix. More precisely, complete ranking of alternatives by the PSI method is based on linear and quadratic transformations of data contained in the decision matrix. Although the PSI method, in essence, belongs to a class of MCDM methods which provide discrete optimization solutions, in this paper an approach based on regression analysis for model development was proposed in order to: (i) model the preference selection index values considering alternative attribute values with regard to the selected criteria in order to analyze the correlations between criteria, (ii) model the preference selection index considering laser cutting parameter values with an aim to predict preference selection index values for any combination of laser cutting parameter values. Regression based modeling of preference selection index values has advantages because
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