Optimization of sequential grinding process in a fuzzy environment using genetic algorithms
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Abstract
The paper presents the methodology of optimization of the sequential grinding process with the application of fuzzy logic for the definition of objectives and constraints imposed on the machining process. The presented method includes the succession of several subsequent operations and the dimensional and shape inaccuracies between them. The use of the fuzzy sets theory enabled the definition of not only the space of expectable solutions, but also the space of acceptable solutions for which the goals and limits imposed on the grinding process are partially met. The presented methodology was used to optimize the process of sequential grinding of small ceramic elements (corundum ceramics with Al_{2}O_{3} content of 92–99%.) The definition of fuzzy objective and constraints in the process of sequential grinding of small ceramics elements was proposed. The influence of the speed of the rotary grinding table and the machining allowance on the deviation of the flatness and height of the grinding elements and the value of the component of the normal grinding force were determined. Using the developed relationships, the definition of fuzzy objectives and constraints defined in the process output parameter space was transferred to the process parameter set space. In such a defined space, the optimization process was carried out using the genetic algorithm. The analysis of the impact of the applied tnorm functions used for aggregation of the fuzzy objective and constraints on the obtained results was performed. It was shown that in the case of sequential grinding of small ceramic elements, the use of minimum tnorm for an aggregation of grinding objective and constraints allows to achieve the highest process efficiency.
Keywords
Grinding processes Dimension and shape accuracy Process efficiency Genetic algorithm Fuzzy optimizationList of symbols
 a_{e}
Total allowance (μm)
 a_{i}
Allowance for the ith grinding wheel (μm)
 C
Process constraint
 D
Diameter of ground elements, mm
 F_{n}
Normal component of the grinding force (N)
 G
Process objective
 Q_{p}
Grinding efficiency (pcs/s)
 Q_{v}
Grinding volume efficiency (mm^{3}/s)
 T_{o}
Distance between consecutive elements on the rotary table perimeter (mm)
 v_{w}
Speed of the grinder rotary table (mm/s)
 V_{w}
Volume of removed material (mm^{3}/pcs)
 X
Set of process input parameters
 y_{acc}
Accepted value of the process output variable
 y_{exp}
Expected value of the process output variable
 Y
Set of process output parameters
 Δh
Workpiece height deviation, μm
 Δp
Workpiece flatness deviation, μm
 μ_{D}
Fuzzy decision
 μ_{Fn}
Degree of fulfillment of the constraint imposed on the normal component of the grinding force
 μ_{Qp}
Degree of fulfillment of the objective imposed on the grinding efficiency
 μ_{Δ h}
Degree of fulfillment of the constraint imposed on the workpiece height deviation
 μ_{Δ p}
Degree of fulfillment of the constraint imposed on the workpiece flatness deviation
1 Introduction
Grinding processes are one of the final operations in the manufacturing process. They determine the accuracy of the shape, dimensions and quality of the machined surface. The result of the grinding process is affected by many factors related to the properties of the grinding tool and workpiece as well as the parameters and conditions of grinding process [1]. The influence of these factors on the grinding results is often characterized by a complex mechanism of their cumulative effects.
In grinding processes, material is removed by abrasive grains located on the grinding wheel active surface. These grains are characterized by a varied shape and random distribution on the surface of the active abrasive tool. The interactions in the contact area of abrasive grain with the workpiece determine the results of the grinding process. Studies carried out in the works [2] indicate the interactions variability in the contact area occurring with the change in grinding parameters. In addition, material separation efficiency also depends on the shape of the abrasive grains [3, 4].
In addition, the topography of the active surface of the abrasive tool changes during grinding as a result of abrasive wear and microcrushing of the surface of abrasive grains and as a result of crushing of whole grains as well as the abrasion of the grinding wheel’s surface with the workpiece. Therefore, the grinding process is characterized by significant randomness.
In the grinding processes of brittle materials, which include aluminum oxide ceramics, two types of material separation can be distinguished: ductile type and brittle type. In the first of these, the material is removed as a result of plastic–elastic material separation. In the second, material removal occurs as a result of cracking and separation of material. Conducting the grinding process with small cutting depth allows the process to be run in the ductile regime, which ensures a higher quality of the treated surface [5, 6].
Selection of parameters and conditions of the grinding process is a frequent demand of manufacturing companies as this operation often corresponds to the final accuracy of the shape and dimensions of the machined elements. The issues concerning optimization of machining processes presented in the literature [7, 8, 9] assume the search for process parameters allowing to obtain: minimum production cost, maximal production rate, finest possible surface quality. Limitations in the grinding process are most often associated with: thermal damage of the ground surface, wear of the grinding wheel, stiffness of the grinderworkpiecegrinding tool system. Determination of the relation of the grinding process parameters on the above values was conducted using analytical methods [10, 11] or neural networks [12, 13]. Gradient methods, nonlinear programming methods and evolutionary algorithms were used to solve optimization tasks [9, 14]. It was demonstrated in the work [15] that the use of evolutionary algorithms to solve the task of grinding processes optimization gives the best results.
The multitude of factors affecting the outcome of the machining process as well as its complexity of mechanisms of cumulation of the effects of their interactions leads to the application of procedures effective in the processing of these types of data. The use of fuzzy logic methods allows for ambiguity and uncertainty in the description of the analyzed phenomena. The paper [16] presents the use of the fuzzy logic methods in the procedure for parameters selection of the surface grinding process. The developed approach enabled the determination of a set of optimal design variables in order to achieve a set of desired process variables. Abbas [17] proposed a method of the optimization under uncertainty in machining processes (abrasive water jet machining, abrasive water jet and ultrasonic machining). The method applied to machining parameters optimization takes into account the variability of the process parameters and their effect on the variability of the machining results. Chiang [18] developed a graybased fuzzy algorithm which simplifies the optimization procedure for the complicated performance characteristics. Rahul et al. [19] described an integrated optimizing path combining satisfaction function, fuzzy inference system and Taguchi method for machining performance optimization for electrodischarge machining of Inconel alloys.
So far, the methods of grinding processes optimization have assumed crisp definitions of objectives and constraints. In this approach, the transition from the set of acceptable processing parameters to the set of unacceptable parameters takes place in a stepwise manner. The variability of the results of grinding process resulting from the randomness of phenomena occurring in the grinding zone is in some contradiction with this approach. A more natural approach would be an introduction of graduation determining the membership of grinding parameters to the set of assumed objectives and constraints. In addition, many optimization criteria are contradictory (e.g., maximum processing efficiency, high quality of processed surface). For a sharp transition from the set of permissible parameters to the set of unacceptable parameters, the degree of fulfillment of the constraints and objective of the grinding process is purely zero–one. For contradictory constraints, when the set of grinding parameters allowing to fulfill all constraints is empty, the lack of graduation in the evaluation of the fulfillment of optimization criteria makes it difficult to evaluate the solutions obtained by optimization.
The introduction of graduation in the description of objectives and constraints cannot be recognized by the application of crisp set theory and enforces the application of fuzzy sets theory [20, 21, 22]. The basics of decision making in a fuzzy environment are described in the study by Belman and Zadeh [23]. Since then, the issues related to decisionmaking methodology in a fuzzy environment have been of an interest to a number of researchers. Tang [24] presents an overview of theories and methods related to fuzzy optimization, classification of fuzzy modeling and optimization, and methods of solving them.
The multitude of factors affecting the result of grinding process and the variability of their effect during the process induces the use of fuzzy logic methods in the definition of machining objectives and constraints. This is particularly important in sequential grinding with the cumulative impact of individual operations inaccuracy on the result of the process.
The paper presents the assumptions and results of the optimization of the grinding process using fuzzy logic to the definition of objectives and constraints imposed on the example of the sequential grinding of small ceramic elements. On the basis of experimental research, the variability of the values of parameters describing the machining accuracy was determined, which was included in the definition of the constraints membership functions. The assumptions of the optimization process also include the sequentiality of the process and the associated fact of propagation of machining inaccuracies between successive machining zones. An objective function has been defined that allows determining the impact of the degree of fulfillment of individual objectives and processing constraints on the result of the fuzzy decision.
Experimental study was carried out to determine the relationships allowing the transfer of objectives and constraints of the grinding process from the output variable space into the grinding parameters space. Process parameters assuring maximization of fuzzy decision using a genetic algorithm were optimized. An analysis of the impact of tnorms used for an aggregation of fuzzy objectives and constraints on optimization results was performed.
2 Basis of making fuzzy decision in fuzzy environment
The objectives and constraints imposed on the machining result mainly from the technological requirements (e.g., the expected value of the surface roughness parameter Sa, the acceptable tolerance of the shape and dimensions) and the economic requirements (e.g., high grinding performance, small grinding cost). In the case of the machining, the membership functions determining the degree of membership of the process output y to a given objective or constraint can be interpreted as functions determining the satisfaction level of the machining result.
The fuzzy decision μ_{D} defines the degree of fulfillment of fuzzy objectives and constraints. The type of tnorm applied significantly affects the aggregated fuzzy decision.
This method provides a selection of machining parameters for maximizing the degree of fulfillment of machining objectives and constraints.
3 Assumptions of the optimization of the process of sequential grinding of small ceramic elements
The method, due to the appropriate kinematics, allows accurate and efficient grinding of ceramic elements. The inclination of the conical grinding wheel increases the grinding path and thus reduces the removal speed of the allowance [25].
During the grinding process, the elements are fixed on the surface of the rotary table; and during the finishing process, they are lifted and flexibly pressed to the active surfaces of the smoothing and polishing wheel.
The assumed objectives in the grinding processes are mainly based on economic requirements and assumptions for assuring the quality requirements of ground parts. The constraints imposed on the grinding process are most often attributed to the properties of the technological devices and the properties of the tool itself, as well as an impact of the grinding parameters on the process quality.
It may be concluded from the above relationship that it is necessary to increase the speed of the grinder rotary table to increase the volume efficiency of the process. Thus, the criterion of the highest speed of the grinder rotary table can be accepted as the ultimate objective of optimization, which is at the same time a controllable parameter of the grinding process.
The selection of grinding parameters should ensure obtaining the assumed grinding efficiency without the negative effects on the dimensions and shape accuracy of the workpieces and the quality of their surface.
The possible range of parameter values depends on the constraints resulting from the properties of the grinder, the workpiece and the grinding wheel. Because the grinding power required to remove the total allowance a_{e} is small in the analyzed grinding process, there is no constraint associated with the power of the grinder. There are constraints related to the accuracy of the shape and dimensions of the workpieces. The machined ceramic elements work as a sliding cooperating part, and proper tightness of connections must be ensured. Accordingly, the processing parameters should ensure the fulfillment of the constraints related to the value of acceptable and expected deviation of height Δh and deviation of flatness Δp of the workpiece. The acceptable and expected value of the parameters results from the requirements for dimension and shape tolerance of the product and the variation of the machining results.
The acceptable and expected values of the output parameters determined on the basis of relationships 12 and 13 may also include expectations regarding the values of the qualitative indicators. For example, for n = 3 and m = 5 the expected value of the C_{pk} capability index will be 1.66 and the permissible value will be 1.
The process of grinding of the brittle ceramic elements is fraught with the risk of fracture microcracking of the ground surface. To ensure a suitable surface quality of the cooperating elements, the restriction of the normal component of the grinding force F_{n} is required.
The functions f_{Δh}, f_{Δp} and f_{Fn} determining the influence of grinding parameters on the process output values were obtained by the experiment described in Sect. 4.
An increase in grinding efficiency (increase in the speed of the rotary table v_{w}) is possible due to appropriate division of the total grinding allowance a_{e}= [a_{1}, a_{2}, a_{3}] among the grinding wheels. The fuzzy constraint of the normal grinding force was imposed on all grinding wheels, whereas the fuzzy constraints of the deviation of height and flatness were imposed on the last grinding wheel in the sequential grinding operation. An assessment of the degree of fulfillment of the fuzzy objective and constraints is made by an aggregation of fuzzy membership functions according to Eq. 6.

minimum: \( t_{\hbox{min} } \left( {a,b} \right) = \hbox{min} \left( {a,b} \right); \);

product: \( t_{\text{prod}} \left( {a,b} \right) = a \cdot b; \);

Łukasiewicz tnorm:
$$ t_{\text{Luk}} \left( {a,b} \right) = \hbox{max} \left( {0,a + b  1} \right); $$ 
Hamacher product:
$$ t_{\text{H}} \left( {a,b} \right) = \left\{ {\begin{array}{*{20}c} 0 & {{\text{if }}a = b = 0} \\ {\frac{a \cdot b}{a + b  a \cdot b}} & {\text{otherwise}} \\ \end{array} } \right.. $$
The grinding process parameters that maximize a fuzzy decision were found using a genetic algorithm. A detailed description of the results of the simulation procedure and their discussion is provided in Sect. 5.
4 Experimental study
Grinding process parameters
Properties  Unit  Value 

Grinding wheel  –  S3020 200 × 85 × 30 × 4 SD 125/100 BT 
Grinding speed, v_{c}  m/s  29.5 m/s 
Allowance, a  μm  50, 100, 200 
Table speed, v_{w}  mm/s  4, 5, 7 
Grinding condition  –  Wet 
Grinding fluid  –  Water 
Mechanical and physical properties of ground elements
Properties  Unit  Value 

Density  g/cm^{3}  3.4–3.8 
Flexural strength  MPa  250–300 
Impact strength  kJ/m^{2}  4 
Emodulus  GPa  220–300 
Hardness (Knoop, 100 g)  GPa  20 
Specific heat capacity  J/kg/K  850 
Thermal conductivity  W/m/K  27.8–34.9 
Due to the fact that the active surface of the grinding wheels is inclined in relation to the plane of the rotary table, a long grinding path has been obtained. The removal speed of the allowance decreases along the grinding path, which is advantageous due to the smaller thermal effects in the machining zone. Water cooling was used in the process due to the fact that in addition to the cooling functions it had a positive effect on the condition of the active surface of the diamond wheels. The active surface retained its ability to last longer, and there was no excessive sticking of the waste products of the machining process.
Workpieces were pressed by the grinding forces to the profiled table edge as a result of the tilting of the grinding wheel at angles α and β (Fig. 2). Cooling with water ensured efficient washing out of the machining products from the clamping zone.
The quality of the ground elements was evaluated by the following devices: LGAGE LG5 laser sensor for height deviation measurement (sensing window range 1.5 mm, sensing beam wavelength 650 nm, response speed slow) and PIK1A analyzer for flatness deviation measurement (sensor type: inductive, measuring tip length 45 mm). During the grinding process, values of the force components were measured using a piezoelectric force sensor with integrated electronics 9602A by Kistler. The measurement data were registered at 10 kHz frequency using a 16bit measurement card.
Influence of grinding parameters on the workpiece height deviation Δh
No.  v_{w} (mm/s)  a (μm)  Δh (μm)  \( \overline{\Delta h} \) (μm)  \( \sigma_{\Delta h} \)  

1  2  3  
1  4  50  4.1  4.4  3.1  3.87  0.68 
2  5  50  10.1  10.2  12.6  10.97  1.31 
3  7  50  14.5  16.1  16.0  15.53  0.84 
4  4  100  7.9  9.1  7.9  8.30  0.63 
5  5  100  16.2  16.4  14.3  15.63  1.10 
6  7  100  21.6  24.8  22.4  22.93  1.67 
7  4  200  16.1  19.2  18.4  17.90  1.62 
8  5  200  25.1  25.7  26.3  25.70  0.63 
9  7  200  34.8  32.7  28.5  32.00  3.30 
\( \bar{\sigma }_{\Delta h} \)  1.31 
Influence of grinding parameters on the workpiece flatness deviation Δp
No.  v_{w} (mm/s)  a (μm)  Δp (μm)  \( \overline{\Delta p} \) (μm)  \( \sigma_{\Delta p} \)  

1  2  3  
1  4  50  1.49  1.73  1.49  1.57  0.13 
2  5  50  1.47  1.62  1.84  1.64  0.19 
3  7  50  2.71  2.73  2.64  2.69  0.05 
4  4  100  1.74  2.10  2.02  1.95  0.19 
5  5  100  2.12  2.46  2.33  2.30  0.18 
6  7  100  3.50  3.42  2.86  3.26  0.33 
7  4  200  3.02  2.77  2.81  2.87  0.13 
8  5  200  4.08  3.81  3.83  3.91  0.14 
9  7  200  4.92  5.51  5.60  5.34  0.36 
\( \bar{\sigma }_{\Delta p} \)  0.19 
Influence of grinding parameters on the normal component of grinding force Fn
No.  v_{w} (mm/s)  a (μm)  F_{n} (N)  \( \bar{F}_{n} \) (N)  \( \sigma_{\text{Fn}} \)  

1  2  3  
1  4  50  15.2  13.8  11.5  13.50  1.94 
2  5  50  15.9  19.2  20.1  18.40  2.20 
3  7  50  25.7  28.4  29.6  27.90  2.04 
4  4  100  20.1  18.9  22.5  20.50  1.88 
5  5  100  28.1  26.2  30.2  28.17  2.09 
6  7  100  33.1  30.9  35.4  33.13  2.35 
7  4  200  35.5  32.8  31.7  33.33  1.99 
8  5  200  33.1  35.0  38.9  35.67  3.03 
9  7  200  35.6  42.1  39.2  38.97  3.40 
\( \bar{\sigma }_{\text{Fn}} \)  2.33 
The analysis of regression coefficients indicates a greater influence of the speed of rotation of the rotary table v_{w} than the value of the allowance a on the value of the monitored output variables (height and flatness deviation as well as normal component of the grinding force). Increasing the value of the allowance, in the analyzed grinding method, increases the length of the grinding zone, which causes that the effect of the increase in the allowance on the normal component of the grinding force is smaller than the influence of the feed speed of the workpiece. The increase in the value of the component of the normal grinding force is one of the main factors affecting the size of the deformation of the machining system, and hence the increase in the deviations of the height and flatness of the workpieces.
The obtained models have a good fit for the experimental data. The average value of determination factor R^{2} for the models is 0.93. Higher impact of the grinder rotary table speed v_{w} on the output value of the process parameters is noticeable in each of the developed models.
5 Fuzzy optimization of process parameters
The range of the initial population values corresponded to the range of input process parameters used in the experiment. A constraintdependent crossover operator was used.

initial population size 500;

generations 20;

function tolerance 1e−10;

constraint tolerance 1e−10.

Case 1: Q_{p}__{acc} = 230 pcs/h, Q_{p_exp} = 286 pcs/h,

Case 2: Q_{p_acc} = 275 pcs/h, Q_{p_exp} = 400 pcs/h.

deviation of height: Δh_{acc} = 10.08 μm, Δh_{exp} = 7.46 μm,

deviation of flatness: Δp_{acc} = 1.84 μm, Δp_{exp} = 1.46 μm,

normal component of grinding force: F_{n}__{acc} = 35 N, F_{n}__{exp} = 30.33 N.
Summary of membership function values for particular fuzzy objective and constraints
tnorm  Value of membership function of fuzzy objectives and constraints  

μΔh  μΔp  μF _{ n1}  μF _{ n2}  μF _{ n3}  μQp  
Case 1  
Minimum  0.94  1.00  0.93  0.93  1.00  0.93 
Product  1.00  1.00  1.00  0.99  1.00  0.85 
Łukasiewicz  1.00  1.00  1.00  1.00  1.00  0.84 
Hamacher  1.00  1.00  0.99  1.00  1.00  0.85 
Case 2  
Minimum  0.38  0.62  0.36  0.36  1.00  0.36 
Product  1.00  1.00  0.66  0.67  1.00  0.17 
Łukasiewicz  No evaluation possibility, \( \mu_{\text{D}} \left( x \right) = 0, \;\;\forall x \in X \)  
Hamacher  0.9  1.00  0.51  0.51  1.00  0.25 
In the case of a large discrepancy between objectives and constraints imposed on the grinding process, the application of the Łukasiewicz tnorm for an aggregation of objectives and fuzzy constraints causes that the value of the fuzzy decision for any values of the grinding parameters equals 0 (Fig. 9c, Table 6). In effect, it is impossible to evaluate the genetic algorithm optimization results.
The results of fuzzy optimization
tnorm  Process parameters x  

v_{w} (mm/s)  a_{1} (μm)  a_{2} (μm)  a_{3} (μm)  
Case 1  
Minimum  4.92  151.16  132.40  16.44 
Product  4.84  150.62  132.74  16.64 
Łukasiewicz  4.84  150.93  132.35  16.73 
Hamacher  4.85  151.36  132.00  16.64 
Case 2  
Minimum  5.59  153.79  131.68  14.53 
Product  5.17  155.28  134.01  10.71 
Łukasiewicz  No evaluation possibility, \( \mu_{\text{D}} \left( x \right) = 0,\;\; \forall x \in X \)  
Hamacher  5.34  155.75  134.64  9.62 
The use of the product tnorm and the Hamacher product tnorm enables to obtain the results in the space between expectation of fulfillment in a full degree of maximum number of fuzzy objectives and constraints (Łukasiewicz tnorm) and the maximization of minimum fulfillment of fuzzy objectives and constraints (minimum tnorm).
In case of the reduced requirements concerning grinding process efficiency, there are no significant technological differences in the values of grinding parameters obtained for different tnorms. The highest efficiency was achieved with the use of the minimum t–norm (for v_{w} = 4.83 mm/s, Q_{p} = 281 pcs/h). An increase in the grinding process efficiency requirements significantly differentiates the optimization results. This is due to the inability of complete fulfillment of the requirements imposed on the process. The best grinding process parameters were obtained when minimum tnorm for aggregation requirements imposed on the grinding process was used. The obtained highest rotary table speed v_{w} = 5.59 mm/s, for which the grinding process efficiency Q_{p} = 319 pcs/h.
6 Summary and conclusions
The stochastic nature of grinding process, caused by the multiplicity of factors affecting its results, induces the use of fuzzy logic methods in the decisionmaking process for the selection of grinding parameters.
The use of classic methods to define objectives and constraints results in the fact that the machining parameters obtained as a result of optimization are often located at the border of acceptable areas. The variability of the grinding process, resulting from the changes of grinding wheel active surface condition, causes the process to pass from the set of admissible parameters into the space of parameters that do not provide the required quality of the process.
Furthermore, the imposition of many restrictions on the machining process, in particular sequential processing, can lead to a situation in which there is no parameter space that meets all the objectives and constraints imposed on the process. (The problem is infeasible.)
In such cases, the application of the fuzzy set theory allows to determine the area of acceptable changes in the value of objectives and constraints imposed on the machining process. It leads to the possibility of assessing the degree of deviation of the optimization results from the expected values.

The application of fuzzy logic to define the fuzzy objectives and constraints allows to consider the degree of fulfillment of the contradictory objectives and constraints imposed on the grinding process during making decisions of the selection of grinding parameters.

The development of models concerning the effect of grinding parameters on the selected output parameters of the grinding process allows to determine the aggregated function of the fuzzy decision in the space of the decision parameters (process input parameters).

The results of the fuzzy optimization are significantly dependent on the tnorm applied to the aggregation of constraints and the objectives of the grinding, and the greater the area of grinding parameters in which the aggregated value of the fuzzy decision takes values different from 0 or 1, the greater the differentiation of the decisions made.

In the case of sequential grinding of small ceramic elements, the use of minimum tnorm for an aggregation of grinding objective and constraints allows to achieve the highest process efficiency.
Notes
Funding
This study was funded by National Science Centre, Poland (Grant # NN 503 557940).
Compliance with ethical standards
Conflict of interest
The authors declare that they have no conflict of interest.
References
 1.Malkin S, Guo C (2008) Grinding technology—theory and applications of machining with abrasives. Industrial Press Inc., New York. ISBN 9780831132477Google Scholar
 2.Hahn RS (1966) On the mechanics of the grinding process under plunge cut conditions. J Eng Ind 88:72–79. https://doi.org/10.1115/1.3670895 CrossRefGoogle Scholar
 3.Rasim A, Mattfeld P, Klocke F (2015) Analysis of the grain shape influence on the chip formation in grinding. J Mater Process Technol 226:60–68. https://doi.org/10.1016/j.jmatprotec.2015.06.041 CrossRefGoogle Scholar
 4.Kacalak W, Lipiński D, Bałasz B, Rypina Ł, Tandecka K, Szafraniec F (2018) Performance evaluation of the grinding wheel with aggregates of grains in grinding of Ti–6Al4 V titanium alloy. Int J Adv Manuf Technol 94:301–314. https://doi.org/10.1007/s001700170905x CrossRefGoogle Scholar
 5.Bifano TG (1988) Ductileregime grinding of brittle materials. Ph.D. thesis, NC State University, Raleigh, NCGoogle Scholar
 6.Yoshioka J, Koizumi K, Shimizu M, Yoshikawa H, Miyashita M, Kanai A (1982) Surface grinding with newly developed ultra precision grinding machine. SME technical paper MR82930Google Scholar
 7.Brinksmeier E, Tönshoff HK, Czenkusch C, Heinzel C (1977) Modelling and optimization of grinding processes. J Intell Manuf 9:303–314. https://doi.org/10.1023/a:1008908724050 CrossRefGoogle Scholar
 8.Álvarez MEP, Bárcena MM, González FA (2016) A review of sustainable machining engineering: optimization process through triple bottom line. J Manuf Sci Eng 138:100801. https://doi.org/10.1115/1.4034277 CrossRefGoogle Scholar
 9.Yusup N, Zain AM, Hashim SZM (2012) Evolutionary techniques in optimizing machining parameters: review and recent applications (2007–2011). Expert Syst Appl 39:9909–9927. https://doi.org/10.1016/j.eswa.2012.02.109 CrossRefGoogle Scholar
 10.Rabiei F, Rahimi AR, Hadad MJ, Ashrafijou A (2015) Performance improvement of minimum quality lubrication (MQL) technique in surface grinding by modeling and optimization. J Clean Prod 86:447–460. https://doi.org/10.1016/j.jclepro.2014.08.045 CrossRefGoogle Scholar
 11.Barrenetxea A, Alvarez J, Marquinez JI, Gallego I, Perello IM, Krajnik P (2016) Stability analysis and optimization algorithms for the setup of infeed centerless grinding. Int J Mach Tools Manuf 84:17–32. https://doi.org/10.1016/j.ijmachtools.2014.04.005 CrossRefGoogle Scholar
 12.Sedighi A, Afshari D (2010) Creep feed grinding optimization by an integrated GANN system. J Intell Manuf 21:657–663. https://doi.org/10.1007/s1084500902434 CrossRefGoogle Scholar
 13.Govindhasamy JJ, Mcloone SF, Irwin GW, French JJ, Doyle RP (2005) Neural modeling, control and optimization of an industrial grinding process. Control Eng Pract 13:1243–1258. https://doi.org/10.1016/j.conengprac.2004.11.006 CrossRefGoogle Scholar
 14.Mukherjee I, Ray PK (2006) A review of optimization techniques in metal cutting processes. Comput Ind Eng 50:15–34. https://doi.org/10.1016/j.cie.2005.10.001 CrossRefGoogle Scholar
 15.Zhang G, Liu M, Li J, Ming WY, Shao XY, Huang Y (2014) Multiobjective optimization for surface grinding process using a hybrid particle swarm optimization algorithm. Int J Adv Manuf Technol 71:1861–1872. https://doi.org/10.1007/s001700135571z CrossRefGoogle Scholar
 16.Chen YT, Kumara SRT (1998) Fuzzy logic and neural networks for design of process parameters: a grinding process application. Int J Prod Res 36:395–415. https://doi.org/10.1080/002075498193804 CrossRefzbMATHGoogle Scholar
 17.Abbas AT, Aly M, Hamza K (2016) Multiobjective optimization under uncertainty in advanced abrasive machining processes via a fuzzyevolutionary approach. J Manuf Sci Eng 138:071003. https://doi.org/10.1115/1.4032567 CrossRefGoogle Scholar
 18.Chiang KT, Liu NM, Chou ChCh (2008) Machining parameters optimization on the die casting process of magnesium alloy using the greybased fuzzy algorithm. Int J Adv Manuf Technol 38:229–237. https://doi.org/10.1007/s001700071103z CrossRefGoogle Scholar
 19.Abhishek K, Datta S, Biswal BB, Mahapatra SS (2017) Machining performance optimization for electrodischarge machining of Inconel 601, 625, 718 and 825: an integrated optimization route combining satisfaction function, fuzzy inference system and Taguchi approach. J Braz Soc Mech Sci Eng 39:3499–3527. https://doi.org/10.1007/s4043001606597 CrossRefGoogle Scholar
 20.Zadeh LA (1965) Fuzzy sets. Inf Control 8:338–353. https://doi.org/10.1016/s00199958(65)90241x CrossRefzbMATHGoogle Scholar
 21.Yager RR (1978) Fuzzy decision making including unequal objectives. Fuzzy Sets Syst 1:87–95. https://doi.org/10.1016/01650114(78)900106 CrossRefzbMATHGoogle Scholar
 22.Kacprzyk J (1997) Multistage fuzzy control: a prescriptive approach. Wiley, New York. ISBN 047196347XzbMATHGoogle Scholar
 23.Bellman RE, Zadeh LA (1970) Decision making in fuzzy environment. Manag Sci 17:B141–B164. https://doi.org/10.1287/mnsc.17.4.b141 MathSciNetCrossRefGoogle Scholar
 24.Tang J, Wang D, Fung RYK, Yung KL (2004) Understanding of fuzzy optimization: theories and methods. J Syst Sci Complex 17:117–136MathSciNetzbMATHGoogle Scholar
 25.Szafraniec F (2018) Innovative methods of surface microgrinding using wheels with a conical and hyperboloid active surface. Ph.D. thesis, Koszalin University of Technology (in Polish)Google Scholar
 26.Duncan AJ (1986) Quality control and industrial statistics, 5th edn. Irwin, Homewood. ISBN 0256035350zbMATHGoogle Scholar
 27.AIAG (2010) Measurement system analysis, reference manual, 4th edn. Chrysler Group LLC, Ford Motor Company, General Motors Corporation. ISBN 9781605342115Google Scholar
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