Development of a software-automated intelligent ...

Development of a software-automated intelligent sculptured surface machining optimization environment Nikolaos A. Fountas, Nikolaos M. Vaxevanidis, Constantinos I. Stergiou & Redha Benhadj-Djilali The International Journal of Advanced Manufacturing Technology ISSN 0268-3768 Int J Adv Manuf Technol DOI 10.1007/s00170-014-6136-5

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Author's personal copy Int J Adv Manuf Technol DOI 10.1007/s00170-014-6136-5

ORIGINAL ARTICLE

Development of a software-automated intelligent sculptured surface machining optimization environment Nikolaos A. Fountas & Nikolaos M. Vaxevanidis & Constantinos I. Stergiou & Redha Benhadj-Djilali

Received: 15 January 2014 / Accepted: 30 June 2014 # Springer-Verlag London 2014

Abstract To ensure the quality of machined products at minimum cost and maximum effectiveness, it is crucial that selection of optimum machining parameters should be done when computer numerically controlled (CNC) machine tools technology is employed. Traditionally, experience of the operator plays a major role in the selection of efficient parameter values; however, attaining optimum ones each time by even skilled end users, is extremely difficult. This paper takes advantage of the possibilities of current computer-aided design (CAD)/computer-aided manufacturing (CAM) technology and implements a genetic algorithm for optimising CNC machining operations mainly for sculptured surfaces. The algorithm has been developed as a hosted application to a cutting-edge CAD/CAM system. Collaboration among applications has been achieved through programming for software automation by utilising the application programme interface of the system. The approach was implemented to a group of test sculptured models with different properties whilst one of them has been actually machined using typical resources. Results obtained after the implementation indicated that the methodology is capable of providing optimum values for process parameters on its way to maintain both productivity and high quality.

Keywords CAM systems . Software automation . Sculptured surfaces . Machining optimization . Artificial intelligence . Genetic algorithms

N. A. Fountas : N. M. Vaxevanidis (*) : C. I. Stergiou : R. Benhadj-Djilali School of Pedagogical and Technological Education (ASPETE), Athens, Greece e-mail: [email protected]

1 Introduction Freeform products are widely used in automotive, aerospace, die/mould and consumer electronics industries. Freeform surfaces are the main characteristic of such products, and they are applied to meet aesthetics and improve functionality. To machine freeform parts, 3- and 5-axis computer numerical control (CNC) machine tools are utilised to remove the unnecessary material from a raw stock and obtain the final surface such that predetermined dimensional accuracy and high quality are achieved. To generate the tool path that a machine tool will follow to produce a freeform part, a computer-aided manufacturing (CAM) system is used for the modelling stage of the individual machining processes (i.e. roughing and finishing). When it comes to machining modelling, the definition of machining strategies and related parameters needs to be done so that the generated tool path will achieve both productivity and quality demands. Tool path generation involves two fundamental aspects, that is, cutting path topology and machining parameters specification. The former deals with the trajectory pattern style the cutter will adapt in order to remove the material whilst the latter involves the selection of suitable values for the parameters to perform the material removal. For a tool path to be characterised as an ideal one, uniform scallops across the machined surface should be left, yet, reducing machining time and maintaining high material removal rate (MRR). The definition of machining parameters for optimum tool path creation has questioned a large number of researchers worldwide. Noticeable studies concerning optimum tool path creation may be distinguished in terms of the milling mode [1, 2], tool geometry and configurations [3], tool positioning [4, 5], machining phases [6] and machining parameters specification [7]. Specification of machining parameters is critical, since it is a part of study for all the aforementioned approaches for tool path creation. In general, the machining parameters available under a tool path strategy are

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treated as independent variables when formulating machining optimization problems. Common machining parameters are spindle speed, feed rate, axial depth of cut (stepdown), radial depth of cut (stepover), machining tolerance, discretization step, scallop height etc. It is mentioned that machining parameters refer to their corresponding machining stage. For sufficient exploration of optimum tool path creation, quality objectives should be determined to reflect problem’s responses. Regarding the machining phase, quality objectives may vary considering the outcomes of both the machined part accuracy and machining efficiency as well. Research works presented so far introduced quality objectives such as minimum machining time (including part, or tool setup time, cutting time, rapid traverse time, time needed for tool changes etc. [8]; maximum MRR; minimum tool deflection [9]; surface roughness; minimum cutting force components [10]; and maximum tool life [11]. Resources employed to machine freeform parts impose the determination of technological constraints so as to ensure feasibility of manufacturing processes. Several constraints are set with variations regarding the scope of optimization. Types of constraints deal with the available spindle power, cutting forces, allowable cutting load and maximum allowable tool deviation [12]; maximum spindle torque [13]; specifications of working intervals for process parameters [14]; cutting tool geometry [15]; tool wear and yielded heat [16, 17]; and geometrical/technological configurations of machine tools stability [18]. Roughing and finishing constitute the most important phases either for prismatic or sculptured surface machining operations. During the roughing process, a high MRR should be achieved. The outcome of rouging operation is a part geometry closed to the ideally designed one. Finishing operation aims at removing the remained material resulting thus to the final product, which should meet predetermined requirements. Roughing is characterised by its ability to increase productivity, whereas high surface quality and dimensional/geometrical accuracy should be the main characteristics of finishing. As a machining sequence, both phases play key roles to a successive manufacturing process. To optimise sculptured surface tool path creation for roughing and/or finishing operations, several strategies have been proposed. Lin and Liu [19] applied an XY-plane interpolator so that point series can be regularly rearranged. Through the application of their approach, the tool path can be automatically generated by taking into account the coordinates from the rearranged point series. Other methodologies involve the usage of cutter location (CL) files to take advantage of the coordinates and develop prediction models so that one or more quality objectives are optimised, thus improving the tool path with reference to them. Towards this direction, De Lacalle et al. in [9] took advantage of such data to integrate cutting force calculations in a commercially available CAM system whilst Tunc and Budak [20] followed similar philosophy to retrieve the machining conditions for process

simulation and analysis. In order to optimise tool path creation in the case of 5-axis surface machining, Gong et al. [21] simulated the machining error in 5-axis machining by applying basic curvature equations of local tool positioning to produce the tool envelope which represents the machined surface. The surface is then compared with the ideal one and results about the deviation are extracted. To do so, a secondorder approximation for the surface envelope is performed, given only one tool position. Manav et al. [22] proposed a new strategy to optimise sculptured surface machining based on mathematical modelling that relates resultant forces, cycle times and scallop heights. Thus, their approach introduces a multi-objective optimization problem to be solved through the objective weighting algorithm. Referring to multi-objective optimization for the machining of sculptured surfaces, a large number of techniques based on artificial intelligence and other related methods has also been employed in several works. Approaches involving intelligence usually implement algorithms to perform loops of experimental evaluations on a given objective function or programming techniques like integer, dynamic and geometric programming [23]. Representative artificial intelligence algorithms are genetic and evolutionary algorithms (GA-EA) [24], simulated annealing (SA) [25], particle swarm optimization (PSO) [26] and artificial neural networks (ANNs) [27]. There is also a tendency to integrate manufacturing software with the abovementioned optimization strategies, since computer-aided tools are strongly recommended to generate NC programmes for complex geometries [28]. Zeroudi et al. [29] managed to establish a computer-assisted methodology to predict cutting forces during machining operations directly from tool path generated by CAM. Their work utilises CL data to compute the tool’s inclination angle relative to the generated surface and then the tool engagement in the material as it was mentioned in [9]. Der Prete et al. [30] proposed a methodology capable of utilising coordinate points measured from a virtual coordinate measuring machine (CMM) so as to optimise machining operations. In contrast to traditional inspection approaches, their work investigates in advance final virtual outputs thus resulting to the reduction of the large machining time required due to the high uncertainty level. Kersting and Zabel [31] implemented the S-metric selection evolutionary multi-objective optimization algorithm to tackle with the problem of sculptured surface machining. Their study emphasises the difficulty of obtaining meaningful, collision-free candidate solutions in terms of tilting and rotational angles in the case of 5-axis milling. Their methodology is integrated to the simulation module of a manufacturing simulation package and case studies are presented to show validity. Antoniadis et al. [32] presented a novel methodology known as milling software needle programme (MSN) to determine resulting surface roughness for ball end milling. It is claimed in their study that accurate machining simulations are

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performed whilst great precision is also achieved regarding tool kinematics.

2 Problem statement and scope of research Current machining optimization based on software is well established in terms of feed rate interpolation to complex surfaces as well as proper tool positioning. That is to say, existing optimization algorithms incorporated to software “learn” exact cutting depths (radial in X-Y level and axial in the Z-level) and angles of cutting trajectories. They are also capable of calculating the material removed by each cut segment. Having such data, the tool motion already determined by end users is divided into smaller segments. Thereby, based on the amount of material removed in each segment, optimum feed rate values for each cutting condition encountered are assigned where necessary and an improved tool path, identical to the original is finally computed. Technically, this approach does not optimise the process under a global scale, since strategies are limited to feed rate or cutting conditions only. Moreover, optimization is applied to only one machining scenario and that is the one formulated utilising human expertise and/or performing trial and error experimentation. It is noted that existing CAM systems include a large number of parameters and practically is the safest way to try several machining scenarios. To avoid partial problem solving, oversimplifications and shortcomings concerning local-scale optimization, this paper proposes a novel optimization methodology based on artificial intelligence with the use of CAM environment. The main contribution of this approach is the automatic value determination for crucial process parameters by an intelligent algorithm, towards its convergence to optimal solutions concerning discrete quality objectives referring to both productivity and product quality. By automating the collaborative environment, process planning time and cost are dramatically reduced whilst final values for machining parameters will lead to optimally formulated tool paths for either prismatic or sculptured surfaces. The methodology investigates both roughing and finishing stages applied to machine parts since series of processes are extensively performed in actual industrial manufacturing operations.

3 Machining modelling using CAM systems 3.1 Current practices Typical CAM software available in the market operates with three-dimensional (3D) models, which represent actual

industrial parts to be machined. With the aid of CAM software, the machining setups are prepared to further apply machining operations. For the machining operations applied to machining setups, cutting strategies along with related process parameters are vital and should be specified. Cutting strategies represent actually tool path style patterns in the form of trajectory lines. Several patterns are available according parts’ geometries and other technical aspects. For roughing, a number of planar slices are defined denoting successive roughing planes, their distance (stepdown) being the same or different, depending on the slope of the boundary surface and the allowable scallop height. Most of roughing strategies apply tool paths consisted of linear parallel segments at a distance from each other that needs to be defined by the user (stepover) for a given machining direction. The way that the tool enters the stock slice to be cut, e.g. plunging, ramping, flank milling etc. is also a matter of choice. In finishing, a small part of the stock is only present and needs to be removed under the constraints of surface finish, which are mainly dealt with through appropriate choice of machining allowance/ tolerance. A variety of 3D strategies may be defined for finishing in exploiting particular shape features of the part. While a number of such strategies just impose a constraint on 3D profiling regarding the direction of the tool path trajectory, e.g. radial, spiral, regarding the shape being machined, some others resulted from a projection of basic planar shapes (line, rectangle, circle, contour, etc.) onto the final part surface coupled again with specific tool path directions. Finally, cutting tools such as end mills, drills, reamers etc., which are represented in the form of solid entities are provided by CAM software through databases. Once machining processes have been modelled, the tool path calculation is conducted according to the predetermined values of the machining parameters. Machining simulation can be then done to check the feasibility of machining processes and extract statistical data such as MRR, machining and/or cycle time, tool path length etc. The traditional workflow of machining modelling in CAM software is depicted in Fig. 1.

3.2 Objectives for quality assessment based on software It is widely accepted that virtual environments lack prediction capabilities when it comes to physical processes as well as their variables. However, it is possible to study quality objectives that involve geometrical attributes to be assessed using machined 3D models and cycle or machining time extracted from simulation experiments. Geometrical attributes may be distinguished regarding the machining stage. This work introduces remaining volume in the form of scallops after roughing and surface deviation after finishing as two major objectives for optimization (Fig. 2).

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Fig. 2 Rough scallops and final surface accuracy of a sculptured model: a ideal design, b stock model, c roughed model and d finished model

Fig. 1 Traditional workflow of machining modelling

3.2.1 Remaining volume on roughed sculptured parts Tool path strategies available for the rough machining of sculptured surfaces result to the excess material in the form of scallops. The excess material is depended on the selection of several attributes such as the tool’s cutting diameter, radial cut engagement (stepover) and axial depth of cut (stepdown). Technically, it is quite difficult to manually select parameter values under a given strategy so that regular and uniform excess material distribution will be achieved. Irregular remaining volume amount inevitably leads to the decision of independently applying rest roughing operations. Rest roughing suggests that cutting tools are to be selected with

end users having sole knowledge of the rest of the tools being used. Thereby, tedious and time-consuming analyses ought to be performed so as to specify the remaining material location. For the reason that no knowledge among tools can be existed, preparing the excess material for its removal by several cutters may impose small diameter tools to remove more material than recommended, whereas larger tools to cut less volume amount than they should. If roughed scallops formulate an irregularly distributed remaining material, cutting force exerted on the tool may not be controlled thus deteriorating final surface, resulting to tool wear phenomena and possibly ending up with tool breakage. To cope with this issue, current industrial practices suggest the selection of smaller tools under less “aggressive” feeds-speeds to formulate uniform roughed surfaces yet; decelerating productivity. Based on the above, the selection of remaining volume as a quality objective to be optimised through software-based environment is meaningful since it introduces an alternative philosophy of examining the possibilities of maintaining constant tool loads for machining operations after roughing. And yet, it can be easily evaluated through software either by comparing models or by calculating it through MRR. The magnitudes integrating this formula are the ones that CAM software handles to compute the tool path. Hence, remaining volume would facilitate software-based optimization functions. 3.2.2 Surface deviation between machined and ideally designed sculptured surfaces The identification of machining error has gained immerse importance for sculptured surface machining. Several research efforts conducted so far concerning the control of machining error especially when experimenting under virtual

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environments so as to ensure safety, reduce scrap and maintain cost at low levels. The selection of surface deviation as a discrete quality objective to evaluate finishing is made so as to observe its significance on machining parameter variations referring to surface quality of machined models. Major scope of its inclusion is the ability of minimising the machining error for later comparisons to actual outputs. It is noted here that the possibility of error compensation when predicting the actual outputs may be investigated, yet it is not under the current work’s prior objectives. Surface deviation represents the difference existing between an ideally designed part surface and the virtually machined one. It is mentioned that this magnitude can also be expressed as the distance between the swept envelope of the cutting tool and the actual design surface as stored in a parametric form in commercial computer-aided design (CAD)/CAM systems [33, 34]. However, this envelope is formulated by processing CL data (CL data) hence it may result to longer computational time and cost. 3.2.3 Computed machining time Machining time is an important objective for all production phases. Referring to machining, it indicates the material removal rate given the volume amount has been removed from the raw stock. Machining time was selected as the second quality objective not only to introduce the trade-off against the two aforementioned objectives but also to orient decision making to comply with the constraints determined as well. 3.3 Manufacturing parameters To successfully formulate a machining optimization problem handled by software, the rationale behind the inclusion of machining parameters to the optimization problem should be such that it satisfies the contradictory nature of the quality objectives. Consecutively, parameters referring to both part geometry (depth of cut, radial engagement etc.) and cycle time (feeds and speeds) should be investigated. The number of machining parameters to be subjected to optimization is not determined by chance but statistically eliminating the less important ones in terms of their influence on these quality objectives, through experimentation. This ensures that only meaningful parameters will be addressed reducing also their computational time. On the other hand, the number of parameters to be investigated should not be small enough in order to avoid over-simplifications and partial problem solving. To decide which of the machining parameters should be involved to optimise quality objectives, Taguchi’s design of experiments (DOE) can be applied as a trustworthy approach to conduct such experiments [2]. Towards the identification of influential manufacturing parameters, one ends up with the ones enlisted in Table 1

according to the machining phase and the tool path strategy selected. Selections made for cutting tools play important role to all individual phases that establish a manufacturing programme for sculptured surface machining. Geometrical configurations of cutting tools should allow accessibility to all part’s features and minimise deflections. Accessibility issues may result to different impact to the remaining volume represented by the scallops from rough cuts whilst deflection occurrence strongly affects machining accuracy and form error magnitude. The tool to be applied for finishing depends on the milling mode. In general, spherical end mills are chosen for 3-axis finishing operations, whereas spherical or flat end mills are selected for a 5-axis simultaneous finish machining. Stepover and stepdown determine the volume to be cut during machining, thus yielding a great impact to tool durability, cutting forces, process efficiency and quality depending on the operation. Technically, large values are applied for roughing to some extent and smaller values are specified to proceed to finishing. Cutting speed and feed rate are dominant parameters to both productivity and machining quality, and their influence to almost all machining objectives studied so far has been verified by numerous research works. Scallop height is differently affected depending on the machining mode. In the case of 3-axis surface machining where spherical end mills are used, scallop height is determined by the radius of the cutter and its radial engagement (stepover) to the work piece material. If 5axis mode is employed, scallop height is affected by tilt angle, which is the tool inclination by a side angle between its vertical axis and the part surface. From a machining quality view, scallop height gives a description of the form error existing on the resulting part, since it influences the number of successive tool paths applied to maintain constant remaining material in each radial cut movement. The values for parameters in the case of simultaneous 5-axis sculptured surface machining ought to fall within certain ranges in order to restrict the freedom from kinematical perspective. Specifically, the two freedom degrees determined by lead and tilt angles result to any position upon the part’s surface resulting to a large number of collision-free contacts, yet thoroughly different. Controlling the range of tool inclination values is of paramount importance so as to avoid tool tip contacts, collisions with either the fixture or the setup and tool deflections.

4 Software-based machining modelling and optimization 4.1 Decision making for optimization approach In general, manufacturing processes are dynamic procedures whose control variables (range of values) vary depending on the specific case. Manufacturing demands for products may be different, and so do manufacturing practices. Under this

Author's personal copy Int J Adv Manuf Technol Table 1 Determination of process parameters depending on the tool path strategy and machining phase Machining phase

Tool path strategy

Process parameters

Symbols, units and short description

Roughing

“Sweep” roughing

Cutting tool

Tool number, Ø (mm), cutting tool for roughing (flat end mill or a special rougher) ae, (%Ø mm), radial tool engagement to the stock material in XY-plane (as a percentage of the selected tool diameter—Ø) ap (mm), depth of cut in z-axis Vc (m/min), the speed of the cutting tool f (mm/rev or mm/min), cutting tool movement in the feed direction Tool number, Ø (mm), cutting tool for roughing (flat end mill or a special rougher) ae (%Ø mm), radial tool engagement to the stock material in XY-plane (as a percentage of the selected tool diameter—Ø) hs (mm), scallop height influences the number of successive tool paths applied to maintain constant remaining material in each radial cut movement Vc (m/min), the speed of the cutting tool f (mm/rev; or mm/min), cutting tool movement in the feed direction Tool number, Ø (mm), selection regarding the geometry (flat end, ball end or corner radius) Vc (m/min), the speed of the cutting tool f (mm/rev; or mm/min), cutting tool movement in the feed direction aL (deg), tool inclination by an angle between its vertical axis and the surface aT (deg), tool inclination by a side angle between its vertical axis and the surface

Stepover

3-axis finishing

Sweeping zig-zag

Stepdown Cutting speed Feed rate Cutting tool Stepover Scallop height

5 axis finishing

Multi axis sweeping zig-zag

Cutting speed Feed rate Tool geometry Cutting speed Feed rate Lead angle Tilt angle

assumption, suitability of an optimization methodology should be tested and evaluated beforehand. For this study, the implementation of GA-EA was deemed to be the most appropriate for the problem against alternative ones owing to their beneficial nature on good adaptation to nonlinear optimization problems such as machining. Strongly based on genetic operators, GA-EA constitute the most pervasive and advanced heuristic techniques in artificial intelligence. Thereby their application against other systems introduces several advantages that previous works have already reported. Compared with similar artificial intelligent techniques, GAEA are by far distinguished for having the ability to simultaneously evaluate many points in the solution domain, thus increasing the probability of reaching global optimum for a given problem. According to research works published so far, their outputs denote that they are both robust and unbiased optimization techniques for sampling a large solution space. Note that GA-EA utilise only a simple scalar performance measure which does not require derivative information. Consecutively, GA-EA are friendly in terms of their implementation. By implementing GA-EA to optimise machining, a good possibility of maintaining both exploration (global search) and exploitation (local search) to the solution space is guaranteed. Exploration is the procedure of visiting thoroughly new regions of the solution domain. Exploitation is the

procedure of visiting these regions of a search domain within the neighbourhood of previously visited regions. That is, exploitation represents the ability of a GA or EA to rapidly search the entire solution space. Under suitable specifications for intelligent operators, GA-EA can achieve a good ratio between exploration and exploitation thus balancing their trade-off and converging to optimum solutions. 4.2 Problem formulation The quality objectives discussed earlier formulate a multiobjective optimization problem whose optimum solutions lie on a Pareto front. When dealing with complex machining operations, the optimization objectives conflict with each other in several regions of this Pareto front. The group of multiple solutions obtained by an intelligent optimization infrastructure may result to an impractical approach in actual procedures performed in workshops and machinery. The weighted method [35] is a practical and well-known strategy to transform a multi-objective optimization problem to a single-objective one. What needs to be determined is the value of a weight that will reflect the significance of the objective that it corresponds to. Thus, the final result will come by the sum of the individually weighted optimization objectives expressed through a convex linear relation. Note

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that the specification of the weights is not standard but depends on the industrial demands referring to the machining optimization problem. Let these individual optimization criteria be OC and their corresponded weights wi ≥0, such that ∑ iwi =1. Hence, a common objective function is generated for evaluation and is written as follows: i X opt wi f i ðocÞ oc w¼1

ð1Þ

where, opt oc wi

fi(oc)

The problem’s optimization type (min or max) The weights assigned to the criteria The individual objective function (if exists) corresponded to each criterion; or the number of optimization criteria.

Following that concept, a machining programme consisting of two phases (i.e. roughing and finishing) formulates an optimization problem that can be mathematically described by the following equations. f ð pr1 ; pr2 ; :::::::prn Þ ¼ minðV R ; t mr Þfor roughing

ð2Þ


f p f 1 ; p f 2 ; … pfn ¼ minðSD; t mf Þfor finishing

ð3Þ

Where, pr1, pr2 … prn VR tmr pf1pf2 … pfm SD tmf

The process parameters involved in roughing The remaining volume considered as a quality objective for roughing The machining time needed for roughing to be concluded The process parameters involved in finishing The surface quality obtained as a quality objective for finishing The machining time needed for finishing to be concluded.

The current study addresses a combinatorial optimization problem consisting of two machining stages; roughing and 3- or 5-axis finishing. To this end, five machining parameters under certain tool path strategies were selected for optimization. The quality objectives of the remaining volume (VR) and rough machining time (t mr ) were selected in the study for roughing; whilst

surface deviation between a finish-machined model and its target model (SD) and finish machining time (tmf) were selected for both 3- and 5-axis finishing operations. Should a manufacturing programme consist of more than these two basic stages, the aforementioned quality objectives may be further extended to adapt to the optimization problem and the quality objectives.

4.3 Development of a genetic-evolutionary algorithm GA-EA have been extensively applied to address noncontinuous optimization problems often encountered in machining. Classical algorithms evolve a standard population size, a random initialization mechanism, a function that performs the iterations to produce new populations and a stopping criterion that usually denotes the maximum number of generations to be reached. Despite the numerous research studies conducted on the implementation of GA-EA to machining optimization, their co-operation with manufacturing software is yet to be investigated. For the development of a GA-EA, an initial chromosome needs to be created with random parameter selections, which are actually the ones to be optimised for each machining phase. That chromosome was chosen to be initialized using binary encoding type [36] through the adaptation of a chromosome matrix, a matrix holding the number of bits and a matrix holding the bit locations and the variables as well (“D-field” matrix; see Table 2). Binary encoding type is preferred owing to its independent nature in terms of real numbers that tend to favour the early found optimal solution when transformed to its phenotypic representation (real value) hence avoiding the inherent bias. The population’s phenotype is calculated as follows: phenð1; jÞ ¼ D

fieldð2; jÞ þ ðD

 sumð1; jÞ

fieldð3; jÞ−D fieldDð2; jÞÞ

ð4Þ

where “sum” denotes the sum of the variable named as “real” and “real” is :
realð1; k Þ ¼ chromð1; k Þ  ð1=2Þbits ð j; kk Þ ð5Þ

4.3.1 Objective functions For both roughing and finishing operations, the objective functions evaluated by the GA were formulated properly so as to represent the multi-objective optimization problem as a single-objective one by assigning weights of influence to the aforementioned criteria.

Author's personal copy Int J Adv Manuf Technol Table 2 Matrices for chromosome calculation for binary encoding “D-field” matrix (bits location and variables’ limits)

Chromosome representation matrix

Number of bits” matrix

3 v1;1 v1;2 :::: v1; j 6⋅ ⋅ ⋅ ⋅ 7 7 IndðiÞ6 4⋅ ⋅ ⋅ ⋅ 5 vi;1 vi;2 ::::vi; j

3 1 2 :::: NumbOfBits 7 6⋅ ⋅ ⋅ ::::: 7 Ind6 5 4⋅ ⋅ ⋅ ::::: 1 2 :::: NumOfBits

2

The resulting weighted sum (WS) chosen according the operation is described via Eq. 6, which also defines

QCðV R ; t mr ; t m f ; S D ; w1 ; w2 ; w3 ; w4 Þ ¼

2

3 BitsLoc UpLim: Low:Lim Ind4 … … … 5 … … …

2

the objective function of GA and computes the performance of candidate solutions in each of the iterations.

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