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Design of robust energy consumption model for manufacturing process considering uncertainties
Journal of Cleaner Production 172 (2018) 119e132
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Journal of Cleaner Production journal homepage: www.elsevier.com/locate/jclepro
Design of robust energy consumption model for manufacturing process considering uncertainties Wei Liao a, Akhil Garg b, Liang Gao a, * a State Key Lab of Digital Manufacturing Equipment & Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, China b Department of Mechatronics Engineering, Shantou University, Shantou, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 May 2017 Received in revised form 14 October 2017 Accepted 14 October 2017
In view of environment degradation, sustainable manufacturing has become a major focus in the production industry. Energy consumption is one of the key factor in sustainable manufacturing and also responsible for an increase in production cost. It is found that machining parameters have a considerable influence on both energy consumption and product quality. One way to optimize the energy consumption is to establish the relationship between the machining parameters. Thus, developing robust and accurate energy consumption models for manufacturing process is an urgent need to ease negative environmental impacts. In this context, an evolutionary approach of Gene Expression Programming considering uncertainties is proposed. Two case studies are carried out to validate the effectiveness of proposed approach. Uncertainties during the modeling process are considered and handled with a designed set of experiments. Experiments are further performed to validate the robustness of the models. Further, 2D and 3D plots are employed to analyze the relationship between the given machining parameters. Optimization of the designed models is then carried out to determine the optimum set of inputs that minimizes the energy consumption. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Energy consumption Face milling Drilling operation Gene expression programming Robust modeling
1. Introduction 1.1. Background and motivations Sustainable manufacturing is paid special attention from the production industry due to the deterioration of the environment and shortage of resources. Energy consumption plays a vital role in sustainable production whose levels are closely related to the environmental impact of manufacturing (Kara and Li, 2011). Moreover, energy is a fundamental resource for any kind of production activity, not to mention that the production industry contributed a total of 31% energy consumption over the world (EIA, 2011). On the other hand, energy consumption also yells a considerable part of production cost. In context of energy saving, great efforts have been put on both electricity distribution such as smart grids development (Amini et al., 2013) and demand side management where technics are employed to reduce energy consumptions (Palensky and Dietrich, 2011). In order to reduce the
* Corresponding author. E-mail address: [email protected] (L. Gao). https://doi.org/10.1016/j.jclepro.2017.10.155 0959-6526/© 2017 Elsevier Ltd. All rights reserved.
production energy costs, a robust and accurate energy consumption model is essential (Peng and Xu, 2014) so that a wide range of optimization methods can be used to fine tune the machining parameters in the model to achieve an optimum balance between production efficiency and energy cost. To achieve this, it is vital to find a reliable modeling approach which performs well in terms of efficiency and generality. The methodologies used for energy consumption modeling are mainly focused on machining theory, empirical formulas, simulation and Artificial Neural Network (ANN). Bayoumi et al. (1994) formulated an energy consumption model in metal cutting based on a closed form mechanistic force model for milling operations. Li and Kara (2011) developed an empirical model for turning operation through power measurements under various cutting conditions. Lu et al. (2016) established a multi-objective multi-pass turning operation model considering both energy consumption and machining quality and applied multi-objective metaheuristics for optimization. Guo et al. (2015) proposed an operation-mode based method for energy consumption prediction of machining processes by performing simulation on material removal mechanism. Kant and Sangwan (2015) developed a face milling energy
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consumption model by utilizing ANN technique and identified the processing time as the main impact factor in milling process. A detailed review of energy consumption modeling can be found in (Zhou et al., 2016). However, interpretable models such as those based on the machining theory lacks accuracy while ANN provides a relative precise and accurate model in a form of black box. The finite element models based on simulation are time consuming and computational intensive. This motivated us to look for a methodology which is capable of developing models having good precision and interpretability. Gene Expression Programming (GEP) is an evolutionary approach introduced by Ferreira (2002) and has been successfully applied for modeling of manufacturing processes. Yang et al. (2013) used GEP for cutting force prediction in face milling. A collaborative evaluation modeling method is employed to evaluate the perfor€ k et al., mance of the obtained model in actual productions. In (Ko 2011), GEP is applied to model the surface roughness of abrasive water jet machining. Experimental results show that the GEP model is in good agreement with actual observations. Yang et al. (2016) proposed a variant of GEP and applied it to model the energy consumption of the milling process. The vital input factors during the milling process are also analyzed. Similarly, Genetic Programming (GP), from where GEP is €log lu derived, has also been applied in manufacturing modeling. Go and Arslan, (2009) used GP for surface roughness prediction in zigzag milling process and compared the GP models with ANN models in terms of accuracy. Milfelner et al. (2005) applied GP in cutting force modeling of milling process and achieved an approximately accuracy of 96.5%. In context of energy conservation, Garg et al. (2015) proposed an energy consumption modeling approach based on GP. In this study, orthogonal basis functions are introduced for model complexity evaluation which has significant influence on the performance of the algorithm. Statistics analysis shows that the accuracy of prediction models can be enhanced by considering model complexity in the objective functions of GP. It is found from the review that the uncertainties occurred during the modeling process are ignored. The settings of the GEP algorithm affect the performance of the modeling process which can be regarded as uncertainties and should be further explored. 1.2. Objectives Face milling and drilling operations are common process frequently used in manufacturing industry. To achieve the goal of sustainable manufacturing, the energy utilization level of these processes should be improved via a reliable and precise model in terms of machining parameters and energy consumption. This study aims to perform modeling and optimization of machining processes to achieve sustainable manufacturing for energy conservation (energy consumption minimization). In this study, GEP will be adopted to formulate the energy consumption models and applied on two case studies on sustainable manufacturing. In the first case study, the main objective is to minimize the energy consumption of face milling process involving machining parameters such as the cutting speed, the depth of cut and the feed rate. To optimize the energy consumption, the traditional trial-and-error procedure may be too costly in some cases (Charoen et al., 2017) and thus makes it more preferable to formulate a robust and precise mathematical model. In this circumstance, the evolutionary approach of GEP is employed to formulate energy consumption model and the 2D and 3D surface analyses are then applied to give in-depth understanding of how machining parameters have impact on the energy efficiency. For the drilling process, temperature, feed and rotational velocity are considered as the inputs to model the mechanical strength and drilling time. Energy consumption is
evaluated indirectly through study of individual factors such as the mechanical strength and operation time. The optimum parameters can be then obtained to minimize the operation time while ensure a satisfactory mechanical strength of the work piece. In actual practice, there exist a few uncertainties in applying GEP. Three main factors, head size, number of generations and selection of objective function are considered as the uncertainties, which have noticeable effect on the performance of algorithm. In order to distinguish how these factors influence the models generated by GEP, multiple comparisons are made with different level of variations. This test shall guide us to tune the algorithm and reduce the uncertainties, thereby, achieving a robust and precise mathematic model. Finally, the best models obtained by GEP can be optimized using various existing optimization methods. Hence, the present work aims to explore the ability of GEP in robust modeling for energy consumption of two manufacturing processes taking the uncertainties into account. Two widely used manufacturing (machining) processes, namely face milling and drilling, are studied to validate proposed methodology. A brief description of problems being considered is shown in Fig. 1. The contributions of this paper may be reflected in energy consumption modeling and experimental analysis through case studies. C A modeling scheme based on evolutionary approach of Gene Expression Programming, which takes uncertainties into account is proposed. Unlike the existing theoretical models, models generated by GEP are explicit, accurate and do not require any pre-assumption of their forms. C Uncertainties during the modeling process are analyzed based on case studies of two widely used machining processes. Different settings of parameters are compared individually and tuned for generation of models with better performance. C Step-by-step experimental studies are carried out to verify the effectiveness of proposed modeling approach. The best models are chosen and further analyzed in terms of performance and validity. The models are then optimized to determine the best parameter settings of the two machining processes that minimize the energy consumption and improve the energy efficiency.
Fig. 1. Illustration of research problems undertaken.
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132
The remainder of the paper is organized as follows. In Section 2, the details of experimental setups for data acquisition for both case studies are discussed. Section 3 introduces the basic concepts of GEP and corresponding uncertainties being investigated. The performance of the GEP models with the uncertainties in the modeling process are also evaluated in Section 3. Simulation design and 2D/ 3D surface analysis for the milling and drilling processes are discussed in Section 4. Finally, Section 5 concludes with recommendations for future work.
2. Experimental setups In this section, two case studies used to validate the proposed methodology are discussed in details. The case studies refer to the two machining processes, whose details including the experimental plan, equipment and assumptions are kept the same as those mentioned in Yang et al. (2016) and Vijayaraghavan et al. (2014). Details of the machining processes that illustrate the nature of collection of data are described as follows.
2.1. Energy consumption data acquisition in face milling operation The face milling operation is performed on a DYNA DM4600 milling machine attached with a Dual Passive Direct Box (DPDB) and corresponding energy consumption is recorded by the power quality analyzer in watts. Cutting tool used in this case is YBC301 SEET12T3-DM, which is a 5 teeth milling cutter with diameter of 100 mm. Cast ZG55 with size of 230 230 108 mm is used as work piece. The whole operation is performed without cutting fluids. Machining parameters being studied here are cutting speed (V, m/min), depth of cut (ap ; mm), feed rate (ft , mm/t) with four different levels, resulting in a total of 64 experiments. Cutting speed describes the relative velocity between the cutting tool and the surface of the work piece, and is determined by spindle speed (n, r/ min) according to Eq. (1). Depth of cut determines the total amount of material removed per pass of the cutting tool. Feed rate is the relative velocity at which the cutter is advanced along the work piece. Table 1 shows the statistical characteristics for the data acquired through the experiments, where kurtosis is a measure of whether the data is heavy-tailed or light-tailed relative to a normal distribution. Details of the dataset can be found in Appendix A. All experiments are carried out in random order to reduce the effects of unknown nuisance variables. The goal of these experiments is to explore how cutting speed, depth of cut and feed rate affects the energy consumption of face milling process.
V¼
pDn 1000
(1)
121
2.2. Mechanical strength and drilling time data acquisition through simulation The outputs being studied in this case are mechanical strength and drilling time considering temperature, feed, rotational velocity as the inputs. The molecular dynamics (MD) simulation is a widelyused technique for theoretical studies on drilling process of graphene, which is a computer simulation method for studying the physical movements of atoms and molecules. In this work, the MD simulation reproduces the process of nanoscale drilling of a graphene work piece. During the simulation process, the covalent bonding of the carbon atoms is described by Brenner's second generation bond order function (REBO) (Brenner et al., 2002) which can be described mathematically as Eq. (2), where bij is the reactive empirical bond order between the atoms, VR and VA are repulsive and attractive pair respectively. The graphene is equilibrated at first to release any residual stresses by achieving thermal equilibrium in an NVT ensemble (Wong and Vijayaraghavan, 2012). In NVT ensemble, the potential energy of the system is minimized by keeping the number of atoms (N), volume (V) and temperature (T) of the system to be constant. The temperature stability of the system is achieved by employing the Nose-Hoover thermostat (Hoover, 1985). Following equilibration, the graphene work piece is subjected to drilling by using a diamond drill bit. The work piece is held in position by fixing the two edges of graphene work piece. The diamond drill bit is initially held at a vertical distance of 20 Å from the graphene work piece. The diamond drill bit is then displaced vertically downwards with a specific feed and rotational velocity to initiate the drilling process. For every 1000 time steps, the system will go through a relax process to ensure atoms are at minimum energy positions. The end atoms are shifted according to the calculation of inward velocity and trajectories of atoms based on Verlet algorithm (Verlet, 1967). The rest of the atoms are also relaxed through the NVT ensemble process until failure occurs. The study on the investigation of the elastic strength of single layer free form graphene sheet is conducted. The elastic response of graphene sheet at temperature, T ¼ 300 K, feed ¼ 10 m/s and rotational velocity ¼ 50 m/s under nanoscale drilling process is noted. It was found that the elastic response of graphene sheet can be divided into three major regions marked as ‘A’, ‘B’ and ‘C’. Region ‘A’ denotes the phase when the drill bit is displaced from its initial position until it makes contact with the graphene sheet. The graphene sheet then begins to deform elastically when the drill bit continues to move further vertically down. This region is marked by the linear increase of the stress of graphene sheet until it reaches a maximum stress of 115 GPa. Each simulation comprises of 500,000 time steps which is equivalent to 500,000 fs. These experiments were performed to investigate the effects of temperature, feed and rotational velocity on the mechanical strength and drilling time for the drilling process. Variations in response due to several combinations of the
Table 1 Experimental data obtained from the milling experiments. Statistical Parameter
Cutting speed (m/min)
Depth of cut (mm)
Feed rate (mm/t)
Energy consumption (W)
Mean Standard error Median Standard deviation Variance Kurtosis Skewness Minimum Maximum
204.193 4.404 204.100 35.231 1241.189 1.370 0.00647 157.000 251.570
1.050 0.0419 1.050 0.335 0.113 1.373 0 0.600 1.500
0.125 0.00699 0.125 0.0559 0.00313 1.373 0 0.0500 0.200
555.672 18.969 529.000 151.750 23028.060 0.592 0.307 274.000 893.000
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inputs are recorded as Appendix B and summarized as Table 2.
EREBO ¼ VR rij bij VA rij
(2)
2.3. Data preparation for the training of models Selection of the training data does affect the performance of modeling approach. Datasets acquired through above experiments are partitioned into two parts through random shuffle, where a random number of samples are picked as the training set with the rest of samples as validation set. In this case, 43 and 42 samples are randomly chosen for the training of models for face milling and drilling processes respectively. The training set is used for formulating the models while the validation set is used for testing the generalization ability of obtained models. 3. Optimization approach of Gene Expression Programming
(a)
(b)
Fig. 2. A typical GEP chromosome (a) Genotype (b) Phenotype.
expressed individually as a sub-tree. The sub-trees are then connected by predefined linking function, such as arithmetic operators like plus, to produce the finally output. That is, the final result is the sum of the output of the sub-trees. Those trees can be easily interpret as mathematical formulations and thus models in tree form are obtained.
3.1. Basic principle and procedure of GEP
t ¼ hðn 1Þ þ 1
(3)
3.1.1. Encoding and decoding scheme Gene expression programming is a variant of Genetic Programming (GP) with advantages of simple fixed length linear encoding from Genetic Algorithm (GA) and capability of representing trees with varying size from GP. Chromosomes of GEP may consists of multiple genes, which is known as multi-gene system, with each gene being divided into two parts, i.e. head and tail. The length of head (h) and tail (t) satisfies Eq. (3), where n is the maximum number of arguments of functions/operators. The head contains symbols from both the function set consisting of arithmetic operators and the terminal set which consists of input variables of the dataset. The tail contains only symbols from the terminal set to ensure that the chromosome always represents a valid tree. Fig. 2 illustrates a typical GEP single-gene chromosome and its corresponding expression tree which is exactly equivalent to Eq. (4), where a and b are input variables. Note that due to the characteristic of GEP in expressing trees of different size with chromosomes of fixed length, some of the coding may not be used in final representation, but such redundancy has been proved to be necessary for a successful evolution (Ferreira, 2006). In this case, the tail and unused coding region are marked in bold and underlined respectively. The genotype and phenotype are terminologies used in genetics. The genotype shows how the chromosome is encoded, while the phenotype illustrates the expression of the chromosome, i.e. the decoding of the chromosome. Generally, GEP uses breadth first decoding scheme to obtain the expression tree of the encoded chromosome. Thus, it is quite straightforward to read the expression trees from left to right and from top to bottom to determine corresponding encodings. As mentioned above, chromosomes may contain multiples genes and each gene can be
f ða; bÞ ¼ a ðb aÞ
(4)
3.1.2. Comparison with GA Since GEP is derived from GA, it follows a similar path of evolutionary process consisting of selection, crossover and mutation. However, there are some fundamental differences between GEP and GA. GEP takes datasets including independent variables such as cutting speed, and response variables like energy consumption as inputs and generates models or programs, in the form of expression trees, to illustrate the relationship between them. Each individual of GEP population stands for a model and therefore the population of GEP is actually a set of models encoded as linear, symbolic strings of fixed length. In brief, GEP is capable of auto generating models from the given datasets. Thus, the main idea of GEP is to search for models which fit the given dataset best while GA searches for optimum solutions, usually composed of a set of numeric variables, for a given problem instance. Consequently, the encoding/decoding and evolutionary operators of GEP are totally different from that of GA's. For a brief view of how the algorithm works, let's take Eq. (2) as example. Assume that the actual expression of Eq. (2) is not known, but we somehow managed to obtain some data about it as in Table 3. The algorithm will firstly initialize the population randomly, decode individuals into expression trees and evaluate corresponding fitness according to the given data. For instance, a chromosome is initialized and decoded as in Fig. 3. Clearly, Fig. 3(b) can be further identified as Eq. (5) (Model A). Take a and b in Table 3
Table 2 Drilling data acquired from MD simulation. Statistical Parameter
Temperature (K)
Feed (m/s)
Rotational velocity (rad/s)
Mechanical strength (GPa)
Time (1012 s)
Mean Standard error Median Standard deviation Variance Kurtosis Skewness Minimum Maximum
370.00 6.41 375.00 43.01 1850.00 0.98 0.41 300.00 425.00
20.00 1.22 20.00 8.16 66.67 1.53 0 10.00 30.00
70.00 2.43 70.00 16.33 266.67 1.53 0 50.00 90.00
75.85 2.37 74.60 15.91 253.12 0.47 0.45 49.65 115.60
74.31 5.54 59.80 37.17 1381.44 1.19 0.62 31.64 147.50
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the training of models while the validation data is for testing and validating the models obtained. 4 Algorithm execution: The fundamental steps of the GEP algorithm can be represented as shown in Fig. 4. GEP starts with a random generated population with every individual representing a model for current dataset. Each model is then evaluated and ranked based on the fitness function. Roulette selection is employed to choose the individuals to be manipulated with genetic operators such as mutation, transposition, crossover/ recombination and so on, forming a new population of the same size. Mutation and crossover here are quite similar to those of GA's with the restriction that only terminal element can appear in the tail of a chromosome. Transposition operator randomly selects a sub-sequence of predefined length from a chromosome and moves it to the head. This loop of evolving new population continues until the stopping criterion is met. The best model of last generation is chosen as the final output of the algorithm.
Table 3 A simple dataset generated from Eq. (2). Input
Output
a
b
1 2 3
1 4 2
(a)
123
0 4 3
(b)
Fig. 3. A GEP chromosome of initialized population (a) Genotype (b) Phenotype.
as inputs and then we can obtain the output of this model/tree as shown in Table 4. If the sum of absolute error of the output is used as the fitness, comparing to the output of the dataset, we can easily calculate the fitness of this model is 14. After evaluation, the chromosomes will go through selection process and then be manipulated by operators to generate offspring. Again, consider the chromosome of Fig. 3, if the mutation operator is applied on the position 2 of the chromosome and randomly changes it from “b” to “-”, resulting in a different model (Model B). It can be found that though the encoding is slightly different from that of Fig. 2(a) in tails, but they do represent the same tree/model, i.e. Eq. (2), which is a perfect match for the dataset in Table 3.
f ða; bÞ ¼ a*b
(5)
3.1.3. Evolving models with Gene Expression Programming A general procedure of implementing GEP in modeling can be described as follows: 1 Definition of the function set and terminal set: The function set consists of arithmetic operators which may be used for modeling while the terminal set includes the independent variables of the dataset. Constants are also regarded as a terminal element which can be random numbers or a set of predefined values. 2 Determination of algorithm parameters and objective/fitness function: Usually, the parameters related to population and number of generation are set by user with the probabilities of evolutionary operators left by default. The objective function is then chosen to compute the fitness of obtained solutions and used to guide the evolutionary search towards the optimum. 3 Data partitioning: The dataset is usually divided into training set and validation set. The training data set is then used by GEP for
Fig. 4. Flowchart of gene expression programming.
Table 4 Output obtained by Model A and Model B. Target value
0 4 3
Model A
Model B
Output
Absolute error
Fitness
Output
Absolute error
Fitness
1 8 6
1 4 9
14
0 4 3
0 0 0
0
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3.2. Uncertainties in Gene Expression Programming The performance of GEP is influenced by the settings of its parameters. Different settings may result in a totally different result. Thus, the settings for GEP are regarded as uncertainties during the modeling process. In this study, three different settings are chosen to find out the influence on the performance of the formulated models, i.e. head size, number of generations and objective function. The objective function plays a vital role in guiding evolution of the population since the fitness is assigned by objective function and individuals with higher fitness are more likely to survive. Therefore, the fitness function greatly impacts the performance of GEP. Usually, many iterations may improve the results but requires more computation resources which influences the effectiveness and efficiency of the modeling process. Consequently, a proper value should be defined to guarantee the convergence while the computation time is constrained. Head size is a unique parameter in GEP which determines the length of the chromosome and a suitable value is needed to ensure that the optimal solution can be encoded within given length while maintaining the proper redundancy for individuals to evolve efficiently. In the following section, comparisons considering the mentioned uncertainties are performed.
3.3. Experimental settings and uncertainties being considered The overall experimental settings are listed as Table 5. Three sets of experiments are designed to investigate how these uncertainties influence the performance of GEP. For each experiment, a total number of 30 independent runs are chosen and the corresponding results are recorded. The statistical results namely average coefficient of determination (R2 ) is used for performance evaluation of the models.
3.3.1. Objective functions Firstly, different objective functions are set for each experiment while keeping other parameters fixed at values as described in Table 5 with number of generations set at 70. The reason for choosing smaller value for iterations is to find out which objective function converges fastest so that the process of modeling can be enhanced. Definitions of considered objective functions are given in Table 6, where RMSEps is a counterpart of RMSE with parsimony pressure. The datasets are partitioned into training set and testing set randomly as mentioned in Section 2. Statistic results of 30 runs are shown as Table 7 with bold numbers highlighting the best R2 obtained for the corresponding case. It is found that both RMSE with or without parsimony pressure outperforms the other two objective functions. Though RMSE and RMSEps have similar performance with RMSE slightly better in terms of accuracy, RMSEps takes program size into consideration and thus can generate formulas with simpler representations. Therefore, RMSEps is chosen as fitness function for further analysis.
3.3.2. Number of generations In context of generations, similarly, parameters are kept the same with RMSEps as fitness function while the only difference between runs is the number of generations. Multiple sets of generations are designed to find a suitable number of iterations for each case, namely to ensure an average R2 of 0.8. For the sake of clarity, only results of generations around best values are given in Table 8. It can be found that, basically, with the increment of number of generations, the R2 also increases which implies that better models are found. However, when the number of generations reaches a certain level, the rate of increase of R2 decreases. Thus, in this case, number of 200, 800 and 1200 generations are large enough for evolving models of high accuracy for the face milling energy, drilling operation strength and operation time
Table 5 Experimental settings for Gene Expression Programming. Function Set
Symbol
Terminal Set
Symbol
Addition Subtraction Multiplication Division Exponential Natural logarithm x to the power of 2
þ e * / Exp Ln
Face milling Cutting speed Depth of cut Feed rate
x1 x2 x3
Sine Cosine General Settings Population size Number of genes Constants per gene Operator Settings Mutation: Fixed-Root Mutation: Function Insertion: Leaf Mutation: Biased Leaf Mutation: Conservative Mutation: Conservative Fixed-Root Mutation: Conservative Function Mutation: Permutation: Conservative Permutation: Biased Mutation: IS Transposition: RIS Transposition: RNC Mutation: Constant Fine-Tuning: Constant Range Finding: Constant Insertion:
x2 Sin Cos
Drilling Temperature
x1
Feed Rotational velocity
x2 x3
30 3 10
Head size Linking Function Constants interval
7 þ [10,10]
0.00138 0.00068 0.00206 0.00546 0.00546 0.00364 0.00182 0.00546 0.00546 0.00546 0.00546 0.00546 0.00546 0.00206 0.00206 0.00009 0.00123
Inversion: Tail Inversion: Tail Mutation: Stumbling Mutation: Uniform Recombination: Uniform Gene Recombination: One-Point Recombination: Two-Point Recombination: Gene Recombination: Gene Transposition: Random Chromosomes: Random Cloning: Best Cloning: Dc Mutation: Dc Inversion: Dc IS Transposition: Dc Permutation:
0.00546 0.00546 0.00546 0.00141 0.00755 0.00755 0.00277 0.00277 0.00277 0.00277 0.0026 0.00102 0.0026 0.00206 0.00546 0.00546 0.00546
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132 Table 6 Objective/Fitness functions being considered. Objective Name (E)
Table 9 Performance comparison of different head size based on R2 .
Definition
Head size
qffiffiffiffiffiffi
RMSE
SSE N
PRESS FPE Fitness Function RMSEps
SSE N
1 þ 2k N
SSE N
Nþk Nk
5 6 7 8 9 10 11
1 $ 1 þ 1 $ kmax k 1000$1þE , 5000 k k max
125
Face milling
Drilling strength
Drilling time
training
testing
training
testing
training
testing
0.789599 0.807719 0.821557 0.816464 0.809168 0.802066
0.911788 0.912991 0.919698 0.913779 0.914333 0.907763
0.654676 0.802785 0.819449 0.784470 0.728971 0.746757
0.602094 0.767343 0.735656 0.680072 0.566847 0.641155
0.759214 0.822761 0.840658 0.807652 0.795476 0.809062
0.722875 0.820694 0.779031 0.789987 0.786481 0.755773
min
where kmax ¼ GðH þ tÞ; kmin ¼ G Others
1 1000$1þE
Note: SSE is sum of squares of error; k is program size of GEP tree; N is number of training samples; kmax and kmin represent maximum and minimum program sizes respectively; G is number of genes.
corresponding to R2 marked in bold are chosen for each case in the following modeling process to achieve the best performance. Finally, through the mentioned experiments, uncertainties in the GEP modeling process are explored. The tuned parameters for each case is summarized as Table 10.
respectively. 4. Simulation design and analysis 3.3.3. Head size It has been found that the head size of chromosome in GEP has a great impact on the algorithm where a too small or too large value may result in poor performance of models. Therefore, the head size is considered as another important uncertainty that should be taken good care of. With parameters set based on the findings of the two prior experiments and Table 5, different head sizes are used to explore the suitable values for each case. Statistical results as shown in Table 9 further validates the previous studies in literature that the relationship between head size and algorithm performance shows a similar trend to a concave upward function (Ferreira, 2006). Especially for the drilling strength case, it is observed that a small head size of 5 causes a significant drop in R2 for both training set and validation set, which implies that the chromosome is too short to encode all the information needed to fully describe the target model. Accordingly, R2 gradually drops if the head size is too large, which brings too much redundancy and causes expansion of search space. Thus, the head sizes
Table 7 Performance comparison of different objective functions based on R2 . Data Set
RMSE RMSEps PRESS FPE
Face milling
Drilling strength
Drilling time
training
testing
training
testing
training
testing
0.754205 0.744335 0.667615 0.603861
0.876609 0.868051 0.808952 0.672498
0.274167 0.243234 0.133083 0.107556
0.218560 0.215868 0.271440 0.294212
0.368709 0.332713 0.115668 0.026080
0.376706 0.360360 0.099850 0.039329
4.1. Model generation and selection In the previous section, the uncertainties of GEP have been explored and an additional 100 runs are performed to acquire the five best models for each case. To validate the obtained models, namely to check if the outputs are within the sensible interval ranges, 8000 inputs samples are generated by assuming it a normal distribution. The statistical results of each model in 3000 validation runs are listed as shown in Table 11. Though, Model 1 obtained from face milling dataset claims the highest R2 , negative values are found in this model which is meaningless since energy consumption should always be positive. With respect to the other four models, no negative output is found and the frequency histograms of these models are illustrated in Fig. 5. Fig. 5 summarizes output values from the GEP models so that their distributions can be easily observed. Since the input values are sampled from normal distributions within certain intervals, it can be expected that the output values of models are more sensible to share similar trends as the inputs. Thus, it can be observed again that a higher R2 does not always imply a better model due to a stronger normal trend is found in Model 5, which has a relatively smaller R2. For the other three models, the outputs are quite centralized compared to Model 5. Therefore, Model 5 is chosen as the final model for energy consumption of face milling process. Similarly, Model 3 and Model 2, which possess good balance between invalid outputs and normal trend, are picked as the best models for drilling mechanical strength and time respectively. Details of chosen models can be found in Appendices. The reason for existence of invalid outputs
Table 8 Performance comparison of different generations based on R2 . Number of Generations
100 200 500 600 700 800 900 1000 1100 1200 1300
Face milling
Drilling time
Drilling strength
training
testing
0.770230 0.820411 0.834009
0.881639 0.930355 0.935569
training
testing
0.639269 0.737960 0.764882 0.815429 0.837152
0.511783 0.720173 0.639295 0.773609 0.815841
training
testing
0.751204 0.765594 0.783571 0.824006 0.847916
0.725675 0.754938 0.789772 0.814897 0.865995
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Table 10 Fine tuned settings for face milling and drilling operation modeling. Data Set
Objective Function
Maximum iterations
Head size
Energy consumption in face milling Drilling strength Drilling time
RMSEps RMSEps RMSEps
500 800 1200
8 7 8
Table 11 Validation check of best five GEP models from 3000 runs. Data Set
Percentage of invalid outputs and training R2 Model 1
Energy consumption in face milling Drilling strength Drilling time
Model 2 (R2 ¼0.890)
2.8025% 0.8273% (R2 ¼0.959) 0.149% (R2 ¼0.964)
(R2 ¼0.870)
0% 0.0539% (R2 ¼0.942) 3.3824% (R2 ¼0.931)
Model 3
Model 4
(R2 ¼0.868)
0% 0.1381% (R2 ¼0.941) 27.994% (R2 ¼0.946)
(R2 ¼0.867)
0% 0.6154% (R2 ¼0.936) 3.081% (R2 ¼0.935)
(a) Model 1
(b) Model 2
(c) Model 3
(d) Model 4
(e) Model 5 Fig. 5. Output distribution of GEP models for energy consumption in face milling.
Model 5 0% (R2 ¼0.867) 0.2959% (R2 ¼0.935) 0.012% (R2 ¼0.931)
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132 Table 12 Prediction values obtained from the best model for face milling energy consumption.
127
Table 14 Prediction values obtained from the best model for drilling time.
Test no.
Target
Prediction
Test no.
Target
Prediction
Test no.
Target
Prediction
Test no.
Target
Prediction
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
311 294 321 274 371 431 393 412 498 482 458 521 754 632 547 839 440 406 365 326 453 406 501 549 535 680 486 625 851 653 773 582
341.93 311.45 372.63 281.21 348.92 493.03 396.67 444.71 558.67 489.34 420.35 628.32 685.22 590.61 496.37 780.21 415.29 385.70 356.33 327.20 449.80 403.74 496.13 542.75 550.48 684.65 483.89 617.40 842.77 659.62 751.01 568.61
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
499 368 431 473 609 456 514 664 620 503 668 516 693 625 785 876 534 403 463 502 652 724 524 590 547 748 650 782 632 893 693 757
458.49 373.33 401.48 429.87 548.09 458.74 503.27 593.18 612.06 741.88 676.81 547.65 729.19 641.13 817.62 906.44 502.29 419.60 446.93 474.50 600.60 644.36 513.92 557.12 611.64 736.90 674.10 800.02 713.93 971.22 799.32 885.08
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
147.50 140.30 135.80 73.80 70.20 65.90 49.20 42.50 37.10 137.11 129.15 123.50 68.39 63.21 59.80 45.93 39.21 35.32 130.65 122.14 116.26 65.17 59.67 56.10 43.69 37.34 34.02 124.50 115.20 108.90 62.30 55.60
126.70 129.98 133.27 76.41 67.40 58.39 59.52 46.76 34.00 117.17 120.46 123.74 71.52 62.50 53.49 56.17 43.41 30.65 111.68 114.96 118.24 68.70 59.69 50.68 54.26 41.49 28.73 105.85 109.13 112.42 65.73 56.72
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
52.50 41.50 35.40 33.10 116.49 107.57 100.75 58.14 52.18 47.19 38.27 33.76 31.64 102.80 95.60 84.30 51.40 45.30 40.80 34.70 45.90 24.60 62.31 50.96 41.73 28.92 24.20 12.46 22.45 16.94 14.13
47.71 52.23 39.47 26.71 99.84 103.12 106.40 62.67 53.66 44.65 50.16 37.40 24.64 82.31 85.59 88.88 53.78 44.77 35.76 44.15 31.39 18.63 66.45 69.73 73.01 45.79 36.78 27.77 38.78 26.02 13.26
Table 13 Prediction values obtained from the best model for drilling mechanical strength. Test no.
Target
Prediction
Test no.
Target
Prediction
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
115.60 102.50 93.30 103.80 92.40 84.90 103.20 92.70 84.50 103.14 85.35 71.12 87.03 76.19 67.98 87.38 77.56 66.22 92.83 77.17 65.03 79.58 71.39 61.28 82.02 69.61 59.20 88.30 71.90 59.10 74.60 64.90
105.91 92.04 86.31 97.97 87.00 82.83 92.93 83.51 80.27 99.41 84.42 77.87 90.15 78.54 73.81 84.27 74.47 70.83 94.99 79.47 72.52 85.06 73.17 68.16 78.76 68.81 64.97 90.19 74.16 66.81 79.60 67.44
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
55.70 74.60 64.90 55.70 81.36 65.24 53.51 59.82 60.78 50.94 70.87 58.32 49.65 64.70 55.40 43.80 51.60 47.10 40.40 37.90 29.50 20.80 52.15 40.17 24.88 42.16 34.81 20.13 39.18 29.81 18.02
62.16 72.88 62.79 58.75 85.22 68.70 60.94 73.97 61.56 56.01 66.83 56.62 52.39 70.75 52.78 43.83 57.52 44.38 38.02 49.11 38.57 33.77 55.20 35.34 24.79 39.32 25.25 17.82 29.24 18.28 12.71
may rely on inefficient training of GEP models to encompass entire domain of the system. However, considering only limited input
values used for training, we can still conclude that the proposed approach performs reasonable well for both the case studies. Finally, comparisons between actual/target values and prediction values obtained by above chosen best models for each case study are shown in Table 12, Tables 13 and 14 respectively. 4.2. Sensitivity analysis To further explore the fundamental effects of each input element, multiple curves (Fig. 6) representing how these elements interact with outputs are plotted by fixing other inputs at mean values. For the face milling process, it can be observed in Fig. 6 (a) that all three inputs, namely depth of cut, feed rate and cutting speed, have considerable influence on energy consumption in a linear monotone increasing manner. Meanwhile, a higher variation in drilling mechanical strength is found with an increase in value of rotational velocity (Fig. 6 (b)), which indicates that rotational velocity is highly sensitive when compared to other two inputs. Fig. 6 (b) also suggests that the mechanical strength decreases nonlinearly with an increase in value of feed and temperature. With respect to drilling time, the relationship between each input factor is shown in Fig. 6 (c). In this case, the temperature becomes the major input element in drilling time model, leading to a higher fluctuation in operation time. Fig. 6 (c) also implies drilling time drops non-linearly with increment in feed and rotational velocity. From the above observation, the operation time can be shortened with trade-off in mechanical strength, states that the efficiency of process usually conflicts with product quality and thus further validates the rationality of GEP models. Similarly, 3D surface plots are used to illustrate the interaction effect of different inputs by fixing one of the inputs at its mean value. It can be observed in Fig. 7 (a) that each individual input has similar effect on energy consumption with combination of feed rate
(a) Energy consumption
(b) Mechanical strength of drilling process
(c) Operation time of drilling process Fig. 6. 2D plots illustrating relationship between inputs and outputs of different models.
(a) Energy consumption in face milling
(b) Mechanical strength of drilling process
(c) Operation time of drilling process Fig. 7. 3D plots illustrating relationship between inputs and outputs of different models.
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132
129
study proposes an evolutionary approach in design of robust energy consumption models considering uncertainties in the experiment and modeling procedures. The main findings from this work are as follows.
Fig. 8. Pareto front found by NSGA-II for drilling models.
and cutting as the dominant inputs. With respect to drilling operation, the combined effect of the rotational velocity and temperature generates higher non-linearity in mechanical strength of drilling process than the combination of other operating parameters. However, the combined effect of the feed and temperature generates higher non-linearity than the combination of other operating parameters in case of operation time. In order to obtain the optimum machining parameters, genetic algorithm is generally employed. However, for energy consumption in face milling, the optimum inputs can be determined by simply setting them as small as possible based on product requirements due to its linearity. The drilling case here is treated as a bi-objective optimization problem and thus NSGA-II, a popular multi-objective metaheuristic proposed by Deb et al. (2002), is used to find the Pareto front for drilling mechanical strength and time. Pareto front is a set of non-dominated solutions, being considered as optimal if no objective can be improved without trade-off at least one objective. Each solution in Pareto front weighs different objectives. In this case, for the drilling operation, decisions can be made by determining a lower boundary for the mechanical strength so that the quality of work piece can be guaranteed and then we may choose the non-dominated solution with minimum processing time and satisfactory mechanical strength. For example, if a minimum strength of 101 GPa is required, then the solution marked with red circle is chosen as shown in Fig. 8. Thus, the operating temperature of 304.72 K, feed rate of 16.87 m/s and rotational velocity of 54.89 rad/s are considered the optimum machining parameters to achieve a satisfactory mechanical strength of 101.08 GPa with a minimum processing time of 65:26 1012 s. In other words, maximum efficiency is achieved and corresponding energy utilization is optimized.
C In this study, prior experiments are carried out to deal with uncertainties of the algorithm, namely selection of objective function, number of iterations and head size determination. Experimental result shows that the objective function and head size have considerable influence on the performance of the algorithm for the given number of iterations. C Validation experiments imply that the GEP models are of good accuracy and performed reasonable well for the two case studies being considered. Models of good accuracy can be successfully obtained within pre-set relatively small iterations. C The behavior of obtained models are further explored using 2D and 3D plots, which reveal the sensitivities of models to different inputs so that the dominant input is determined. Through these graphs, the fundamental relationship between all the inputs can be clearly seen and thus can be used as guidance for actual productions and model validation. Furthermore, metaheuristics are employed to search for optimum parameters for the model in achieving lower energy consumption for both case studies. C The models formulated (Appendices) are explicit and can be coded into the system for online monitoring. The models can also be used in shop-floor of the production industry by workers/managers. Meanwhile, the present study possesses some limitations which can be treated as future research directions. To further improve the training of GEP models in case of limited data samples, Monte Carlo sampling technique can be integrated in framework of GEP to generate the moderate size data. Thus, the robustness and accuracy of model can be further enhanced.
Acknowledgments This research work was supported by National Natural Science Foundation of China (NSFC) under Grant nos. 51435009, Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant nos. 51421062 and Shantou University Scientific Research Foundation (NTF 16002).
Appendices 5. Conclusions Appendix A Sustainable manufacturing plays a vital role in production industry due to the prominent problem of environment. The present
Table A.1 Dataset obtained from face milling experiments. Test no.
V(m/min)
ap (mm)
ft (mm/t)
EC(w)
Test no.
V(m/min)
ap (mm)
ft (mm/t)
EC(w)
1 2 3 4 5 6 7
157 157 157 157 188.4 188.4 188.4
0.6 0.6 0.6 0.6 0.6 0.6 0.6
0.15 0.1 0.2 0.05 0.05 0.2 0.1
311 294 321 274 371 431 393
33 34 35 36 37 38 39
157 157 157 157 188.4 188.4 188.4
1.2 1.2 1.2 1.2 1.2 1.2 1.2
0.2 0.05 0.1 0.15 0.15 0.05 0.1
499 368 431 473 609 456 514
(continued on next page)
130
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132
Table A.1 (continued ) Test no.
V(m/min)
ap (mm)
ft (mm/t)
EC(w)
Test no.
V(m/min)
ap (mm)
ft (mm/t)
EC(w)
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
188.4 219.8 219.8 219.8 219.8 251.57 251.57 251.57 251.57 157 157 157 157 188.4 188.4 188.4 188.4 219.8 219.8 219.8 219.8 251.57 251.57 251.57 251.57
0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9
0.15 0.15 0.1 0.05 0.2 0.15 0.1 0.05 0.2 0.2 0.15 0.1 0.05 0.1 0.05 0.15 0.2 0.1 0.2 0.05 0.15 0.2 0.1 0.15 0.05
412 498 482 458 521 754 632 547 839 440 406 365 326 453 406 501 549 535 680 486 625 851 653 773 582
40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
188.4 219.8 219.8 219.8 219.8 251.57 251.57 251.57 251.57 157 157 157 157 188.4 188.4 188.4 188.4 219.8 219.8 219.8 219.8 251.57 251.57 251.57 251.57
1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
0.2 0.1 0.2 0.15 0.05 0.1 0.05 0.15 0.2 0.2 0.05 0.1 0.15 0.15 0.2 0.05 0.1 0.05 0.15 0.1 0.2 0.05 0.2 0.1 0.15
664 620 503 668 516 693 625 785 876 534 403 463 502 652 724 524 590 547 748 650 782 632 893 693 757
Note: V represents cutting speed, ap is depth of cut, ft is feed rate and EC stands for energy consumption.
Appendix B
Table B.1 Dataset obtained from face milling experiments. Test no
Temperature (K)
Feed (m/s)
Rotation Speed (rad/s)
Strength (Gpa)
Time (1012 s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
300 300 300 300 300 300 300 300 300 350 350 350 350 350 350 350 350 350 375 375 375 375 375 375 375 375 375 400 400 400 400 400 400 400 400 400 425
10 10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30 10
50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50
115.6 102.5 93.3 103.8 92.4 84.9 103.2 92.7 84.5 103.14 85.35 71.12 87.03 76.19 67.98 87.38 77.56 66.22 92.83 77.17 65.03 79.58 71.39 61.28 82.02 69.61 59.2 88.3 71.9 59.1 74.6 64.9 55.7 74.6 64.9 55.7 81.36
147.5 140.3 135.8 73.8 70.2 65.9 49.2 42.5 37.1 137.11 129.15 123.5 68.39 63.21 59.8 45.93 39.21 35.32 130.65 122.14 116.26 65.17 59.67 56.1 43.69 37.34 34.02 124.5 115.2 108.9 62.3 55.6 52.5 41.5 35.4 33.1 116.49
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132
131
Table B.1 (continued ) Test no
Temperature (K)
Feed (m/s)
Rotation Speed (rad/s)
Strength (Gpa)
Time (1012 s)
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
425 425 425 425 425 425 425 425 500 500 500 500 500 500 500 500 500 600 600 600 600 600 600 600 600 600
10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30
70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90
65.24 53.51 59.82 60.78 50.94 70.87 58.32 49.65 64.7 55.4 43.8 51.6 47.1 40.4 37.9 29.5 20.8 52.15 40.17 24.88 42.16 34.81 20.13 39.18 29.81 18.02
107.57 100.75 58.14 52.18 47.19 38.27 33.76 31.64 102.8 95.6 84.3 51.4 45.3 40.8 34.7 45.9 24.6 62.31 50.96 41.73 28.92 24.2 12.46 22.45 16.94 14.13
Appendix C
Energy consumption ¼ ðdð1Þdð2ÞÞ þ ððððdð1Þ þ dð2ÞÞ=ðG2C5 dð2ÞÞÞ ððG2C2G2C9Þ þ ðdð3Þ þ G2C5ÞÞÞððG2C5
G2C7 ¼ 5.67190810876797, G2C6 ¼ 7.09601123081149, G2C1 ¼ 0.408642841883602, G3C2 ¼ 0.777916806543168, G3C4 ¼ 5.27279915768914, G3C0 ¼ 0.573442793053987, d(1) is temperature, d(2) is feed and d(3) is rotational velocity.
þ dð1ÞÞðdð3Þ=G2C9ÞÞÞ þ ððððdð1Þ þ expðdð2ÞÞÞðlnðdð1ÞÞdð3ÞÞÞ þ expð
Appendix E
ðdð2Þ þ G3C1ÞÞÞ þ dð1ÞÞ (C.1)
where.
G2C5 ¼ 8.26105533005768, G2C9 ¼ 3.21284127323222, G2C2 ¼ 4.34369945371868, G3C1 ¼ 8.28012939848018, d(1) is cutting speed, d(2) is depth of cut and d(3) is feed rate.
Drilling time ¼ ðððdð2ÞG1C1Þ ðG1C2G1C8ÞÞ ððdð3Þ ^ G1C8Þ cosðdð1ÞÞÞÞ þ exp G2C42 . þ ððdð3Þ G2C3ÞG2C5Þ dð2Þ þ ð ðcosðdð1ÞÞðdð1Þ=dð2ÞÞÞ þ ðdð3Þ=ðG3C3 þ dð2ÞÞÞÞ where.
(E.1)
Appendix D
Drilling strength ¼
G1C1 ¼ 0.148716696676534, G1C2 ¼ 9.87925016022217, G1C8 ¼ 9.44456312753685, G2C5 ¼ 9.66063417462691, G2C4 ¼ 2.37126682332835, G2C3 ¼ 9.4550371929075, G3C3 ¼ 4.95236249732963, d(1) is temperature, d(2) is feed and d(3) is rotational velocity.
^ sin ln dð1Þ2 ðsinðdð3ÞÞ ðG1C3G1C3ÞÞ þ ðððG2C7dð1ÞÞ =cosðG2C6ÞÞ=ððdð2Þ þ dð3ÞÞ þ ðdð2Þ þ G2C1ÞÞÞ þ ð ðsinðdð3ÞÞsinðG3C2ÞÞðsinðdð1ÞÞ
References
þ ðG3C4 þ G3C0ÞÞÞ where. G1C3 ¼ 8.66756187627796,
(D.1)
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Contents lists available at ScienceDirect
Journal of Cleaner Production journal homepage: www.elsevier.com/locate/jclepro
Design of robust energy consumption model for manufacturing process considering uncertainties Wei Liao a, Akhil Garg b, Liang Gao a, * a State Key Lab of Digital Manufacturing Equipment & Technology, School of Mechanical Science and Engineering, Huazhong University of Science and Technology, Wuhan, China b Department of Mechatronics Engineering, Shantou University, Shantou, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 23 May 2017 Received in revised form 14 October 2017 Accepted 14 October 2017
In view of environment degradation, sustainable manufacturing has become a major focus in the production industry. Energy consumption is one of the key factor in sustainable manufacturing and also responsible for an increase in production cost. It is found that machining parameters have a considerable influence on both energy consumption and product quality. One way to optimize the energy consumption is to establish the relationship between the machining parameters. Thus, developing robust and accurate energy consumption models for manufacturing process is an urgent need to ease negative environmental impacts. In this context, an evolutionary approach of Gene Expression Programming considering uncertainties is proposed. Two case studies are carried out to validate the effectiveness of proposed approach. Uncertainties during the modeling process are considered and handled with a designed set of experiments. Experiments are further performed to validate the robustness of the models. Further, 2D and 3D plots are employed to analyze the relationship between the given machining parameters. Optimization of the designed models is then carried out to determine the optimum set of inputs that minimizes the energy consumption. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Energy consumption Face milling Drilling operation Gene expression programming Robust modeling
1. Introduction 1.1. Background and motivations Sustainable manufacturing is paid special attention from the production industry due to the deterioration of the environment and shortage of resources. Energy consumption plays a vital role in sustainable production whose levels are closely related to the environmental impact of manufacturing (Kara and Li, 2011). Moreover, energy is a fundamental resource for any kind of production activity, not to mention that the production industry contributed a total of 31% energy consumption over the world (EIA, 2011). On the other hand, energy consumption also yells a considerable part of production cost. In context of energy saving, great efforts have been put on both electricity distribution such as smart grids development (Amini et al., 2013) and demand side management where technics are employed to reduce energy consumptions (Palensky and Dietrich, 2011). In order to reduce the
* Corresponding author. E-mail address: [email protected] (L. Gao). https://doi.org/10.1016/j.jclepro.2017.10.155 0959-6526/© 2017 Elsevier Ltd. All rights reserved.
production energy costs, a robust and accurate energy consumption model is essential (Peng and Xu, 2014) so that a wide range of optimization methods can be used to fine tune the machining parameters in the model to achieve an optimum balance between production efficiency and energy cost. To achieve this, it is vital to find a reliable modeling approach which performs well in terms of efficiency and generality. The methodologies used for energy consumption modeling are mainly focused on machining theory, empirical formulas, simulation and Artificial Neural Network (ANN). Bayoumi et al. (1994) formulated an energy consumption model in metal cutting based on a closed form mechanistic force model for milling operations. Li and Kara (2011) developed an empirical model for turning operation through power measurements under various cutting conditions. Lu et al. (2016) established a multi-objective multi-pass turning operation model considering both energy consumption and machining quality and applied multi-objective metaheuristics for optimization. Guo et al. (2015) proposed an operation-mode based method for energy consumption prediction of machining processes by performing simulation on material removal mechanism. Kant and Sangwan (2015) developed a face milling energy
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consumption model by utilizing ANN technique and identified the processing time as the main impact factor in milling process. A detailed review of energy consumption modeling can be found in (Zhou et al., 2016). However, interpretable models such as those based on the machining theory lacks accuracy while ANN provides a relative precise and accurate model in a form of black box. The finite element models based on simulation are time consuming and computational intensive. This motivated us to look for a methodology which is capable of developing models having good precision and interpretability. Gene Expression Programming (GEP) is an evolutionary approach introduced by Ferreira (2002) and has been successfully applied for modeling of manufacturing processes. Yang et al. (2013) used GEP for cutting force prediction in face milling. A collaborative evaluation modeling method is employed to evaluate the perfor€ k et al., mance of the obtained model in actual productions. In (Ko 2011), GEP is applied to model the surface roughness of abrasive water jet machining. Experimental results show that the GEP model is in good agreement with actual observations. Yang et al. (2016) proposed a variant of GEP and applied it to model the energy consumption of the milling process. The vital input factors during the milling process are also analyzed. Similarly, Genetic Programming (GP), from where GEP is €log lu derived, has also been applied in manufacturing modeling. Go and Arslan, (2009) used GP for surface roughness prediction in zigzag milling process and compared the GP models with ANN models in terms of accuracy. Milfelner et al. (2005) applied GP in cutting force modeling of milling process and achieved an approximately accuracy of 96.5%. In context of energy conservation, Garg et al. (2015) proposed an energy consumption modeling approach based on GP. In this study, orthogonal basis functions are introduced for model complexity evaluation which has significant influence on the performance of the algorithm. Statistics analysis shows that the accuracy of prediction models can be enhanced by considering model complexity in the objective functions of GP. It is found from the review that the uncertainties occurred during the modeling process are ignored. The settings of the GEP algorithm affect the performance of the modeling process which can be regarded as uncertainties and should be further explored. 1.2. Objectives Face milling and drilling operations are common process frequently used in manufacturing industry. To achieve the goal of sustainable manufacturing, the energy utilization level of these processes should be improved via a reliable and precise model in terms of machining parameters and energy consumption. This study aims to perform modeling and optimization of machining processes to achieve sustainable manufacturing for energy conservation (energy consumption minimization). In this study, GEP will be adopted to formulate the energy consumption models and applied on two case studies on sustainable manufacturing. In the first case study, the main objective is to minimize the energy consumption of face milling process involving machining parameters such as the cutting speed, the depth of cut and the feed rate. To optimize the energy consumption, the traditional trial-and-error procedure may be too costly in some cases (Charoen et al., 2017) and thus makes it more preferable to formulate a robust and precise mathematical model. In this circumstance, the evolutionary approach of GEP is employed to formulate energy consumption model and the 2D and 3D surface analyses are then applied to give in-depth understanding of how machining parameters have impact on the energy efficiency. For the drilling process, temperature, feed and rotational velocity are considered as the inputs to model the mechanical strength and drilling time. Energy consumption is
evaluated indirectly through study of individual factors such as the mechanical strength and operation time. The optimum parameters can be then obtained to minimize the operation time while ensure a satisfactory mechanical strength of the work piece. In actual practice, there exist a few uncertainties in applying GEP. Three main factors, head size, number of generations and selection of objective function are considered as the uncertainties, which have noticeable effect on the performance of algorithm. In order to distinguish how these factors influence the models generated by GEP, multiple comparisons are made with different level of variations. This test shall guide us to tune the algorithm and reduce the uncertainties, thereby, achieving a robust and precise mathematic model. Finally, the best models obtained by GEP can be optimized using various existing optimization methods. Hence, the present work aims to explore the ability of GEP in robust modeling for energy consumption of two manufacturing processes taking the uncertainties into account. Two widely used manufacturing (machining) processes, namely face milling and drilling, are studied to validate proposed methodology. A brief description of problems being considered is shown in Fig. 1. The contributions of this paper may be reflected in energy consumption modeling and experimental analysis through case studies. C A modeling scheme based on evolutionary approach of Gene Expression Programming, which takes uncertainties into account is proposed. Unlike the existing theoretical models, models generated by GEP are explicit, accurate and do not require any pre-assumption of their forms. C Uncertainties during the modeling process are analyzed based on case studies of two widely used machining processes. Different settings of parameters are compared individually and tuned for generation of models with better performance. C Step-by-step experimental studies are carried out to verify the effectiveness of proposed modeling approach. The best models are chosen and further analyzed in terms of performance and validity. The models are then optimized to determine the best parameter settings of the two machining processes that minimize the energy consumption and improve the energy efficiency.
Fig. 1. Illustration of research problems undertaken.
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132
The remainder of the paper is organized as follows. In Section 2, the details of experimental setups for data acquisition for both case studies are discussed. Section 3 introduces the basic concepts of GEP and corresponding uncertainties being investigated. The performance of the GEP models with the uncertainties in the modeling process are also evaluated in Section 3. Simulation design and 2D/ 3D surface analysis for the milling and drilling processes are discussed in Section 4. Finally, Section 5 concludes with recommendations for future work.
2. Experimental setups In this section, two case studies used to validate the proposed methodology are discussed in details. The case studies refer to the two machining processes, whose details including the experimental plan, equipment and assumptions are kept the same as those mentioned in Yang et al. (2016) and Vijayaraghavan et al. (2014). Details of the machining processes that illustrate the nature of collection of data are described as follows.
2.1. Energy consumption data acquisition in face milling operation The face milling operation is performed on a DYNA DM4600 milling machine attached with a Dual Passive Direct Box (DPDB) and corresponding energy consumption is recorded by the power quality analyzer in watts. Cutting tool used in this case is YBC301 SEET12T3-DM, which is a 5 teeth milling cutter with diameter of 100 mm. Cast ZG55 with size of 230 230 108 mm is used as work piece. The whole operation is performed without cutting fluids. Machining parameters being studied here are cutting speed (V, m/min), depth of cut (ap ; mm), feed rate (ft , mm/t) with four different levels, resulting in a total of 64 experiments. Cutting speed describes the relative velocity between the cutting tool and the surface of the work piece, and is determined by spindle speed (n, r/ min) according to Eq. (1). Depth of cut determines the total amount of material removed per pass of the cutting tool. Feed rate is the relative velocity at which the cutter is advanced along the work piece. Table 1 shows the statistical characteristics for the data acquired through the experiments, where kurtosis is a measure of whether the data is heavy-tailed or light-tailed relative to a normal distribution. Details of the dataset can be found in Appendix A. All experiments are carried out in random order to reduce the effects of unknown nuisance variables. The goal of these experiments is to explore how cutting speed, depth of cut and feed rate affects the energy consumption of face milling process.
V¼
pDn 1000
(1)
121
2.2. Mechanical strength and drilling time data acquisition through simulation The outputs being studied in this case are mechanical strength and drilling time considering temperature, feed, rotational velocity as the inputs. The molecular dynamics (MD) simulation is a widelyused technique for theoretical studies on drilling process of graphene, which is a computer simulation method for studying the physical movements of atoms and molecules. In this work, the MD simulation reproduces the process of nanoscale drilling of a graphene work piece. During the simulation process, the covalent bonding of the carbon atoms is described by Brenner's second generation bond order function (REBO) (Brenner et al., 2002) which can be described mathematically as Eq. (2), where bij is the reactive empirical bond order between the atoms, VR and VA are repulsive and attractive pair respectively. The graphene is equilibrated at first to release any residual stresses by achieving thermal equilibrium in an NVT ensemble (Wong and Vijayaraghavan, 2012). In NVT ensemble, the potential energy of the system is minimized by keeping the number of atoms (N), volume (V) and temperature (T) of the system to be constant. The temperature stability of the system is achieved by employing the Nose-Hoover thermostat (Hoover, 1985). Following equilibration, the graphene work piece is subjected to drilling by using a diamond drill bit. The work piece is held in position by fixing the two edges of graphene work piece. The diamond drill bit is initially held at a vertical distance of 20 Å from the graphene work piece. The diamond drill bit is then displaced vertically downwards with a specific feed and rotational velocity to initiate the drilling process. For every 1000 time steps, the system will go through a relax process to ensure atoms are at minimum energy positions. The end atoms are shifted according to the calculation of inward velocity and trajectories of atoms based on Verlet algorithm (Verlet, 1967). The rest of the atoms are also relaxed through the NVT ensemble process until failure occurs. The study on the investigation of the elastic strength of single layer free form graphene sheet is conducted. The elastic response of graphene sheet at temperature, T ¼ 300 K, feed ¼ 10 m/s and rotational velocity ¼ 50 m/s under nanoscale drilling process is noted. It was found that the elastic response of graphene sheet can be divided into three major regions marked as ‘A’, ‘B’ and ‘C’. Region ‘A’ denotes the phase when the drill bit is displaced from its initial position until it makes contact with the graphene sheet. The graphene sheet then begins to deform elastically when the drill bit continues to move further vertically down. This region is marked by the linear increase of the stress of graphene sheet until it reaches a maximum stress of 115 GPa. Each simulation comprises of 500,000 time steps which is equivalent to 500,000 fs. These experiments were performed to investigate the effects of temperature, feed and rotational velocity on the mechanical strength and drilling time for the drilling process. Variations in response due to several combinations of the
Table 1 Experimental data obtained from the milling experiments. Statistical Parameter
Cutting speed (m/min)
Depth of cut (mm)
Feed rate (mm/t)
Energy consumption (W)
Mean Standard error Median Standard deviation Variance Kurtosis Skewness Minimum Maximum
204.193 4.404 204.100 35.231 1241.189 1.370 0.00647 157.000 251.570
1.050 0.0419 1.050 0.335 0.113 1.373 0 0.600 1.500
0.125 0.00699 0.125 0.0559 0.00313 1.373 0 0.0500 0.200
555.672 18.969 529.000 151.750 23028.060 0.592 0.307 274.000 893.000
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inputs are recorded as Appendix B and summarized as Table 2.
EREBO ¼ VR rij bij VA rij
(2)
2.3. Data preparation for the training of models Selection of the training data does affect the performance of modeling approach. Datasets acquired through above experiments are partitioned into two parts through random shuffle, where a random number of samples are picked as the training set with the rest of samples as validation set. In this case, 43 and 42 samples are randomly chosen for the training of models for face milling and drilling processes respectively. The training set is used for formulating the models while the validation set is used for testing the generalization ability of obtained models. 3. Optimization approach of Gene Expression Programming
(a)
(b)
Fig. 2. A typical GEP chromosome (a) Genotype (b) Phenotype.
expressed individually as a sub-tree. The sub-trees are then connected by predefined linking function, such as arithmetic operators like plus, to produce the finally output. That is, the final result is the sum of the output of the sub-trees. Those trees can be easily interpret as mathematical formulations and thus models in tree form are obtained.
3.1. Basic principle and procedure of GEP
t ¼ hðn 1Þ þ 1
(3)
3.1.1. Encoding and decoding scheme Gene expression programming is a variant of Genetic Programming (GP) with advantages of simple fixed length linear encoding from Genetic Algorithm (GA) and capability of representing trees with varying size from GP. Chromosomes of GEP may consists of multiple genes, which is known as multi-gene system, with each gene being divided into two parts, i.e. head and tail. The length of head (h) and tail (t) satisfies Eq. (3), where n is the maximum number of arguments of functions/operators. The head contains symbols from both the function set consisting of arithmetic operators and the terminal set which consists of input variables of the dataset. The tail contains only symbols from the terminal set to ensure that the chromosome always represents a valid tree. Fig. 2 illustrates a typical GEP single-gene chromosome and its corresponding expression tree which is exactly equivalent to Eq. (4), where a and b are input variables. Note that due to the characteristic of GEP in expressing trees of different size with chromosomes of fixed length, some of the coding may not be used in final representation, but such redundancy has been proved to be necessary for a successful evolution (Ferreira, 2006). In this case, the tail and unused coding region are marked in bold and underlined respectively. The genotype and phenotype are terminologies used in genetics. The genotype shows how the chromosome is encoded, while the phenotype illustrates the expression of the chromosome, i.e. the decoding of the chromosome. Generally, GEP uses breadth first decoding scheme to obtain the expression tree of the encoded chromosome. Thus, it is quite straightforward to read the expression trees from left to right and from top to bottom to determine corresponding encodings. As mentioned above, chromosomes may contain multiples genes and each gene can be
f ða; bÞ ¼ a ðb aÞ
(4)
3.1.2. Comparison with GA Since GEP is derived from GA, it follows a similar path of evolutionary process consisting of selection, crossover and mutation. However, there are some fundamental differences between GEP and GA. GEP takes datasets including independent variables such as cutting speed, and response variables like energy consumption as inputs and generates models or programs, in the form of expression trees, to illustrate the relationship between them. Each individual of GEP population stands for a model and therefore the population of GEP is actually a set of models encoded as linear, symbolic strings of fixed length. In brief, GEP is capable of auto generating models from the given datasets. Thus, the main idea of GEP is to search for models which fit the given dataset best while GA searches for optimum solutions, usually composed of a set of numeric variables, for a given problem instance. Consequently, the encoding/decoding and evolutionary operators of GEP are totally different from that of GA's. For a brief view of how the algorithm works, let's take Eq. (2) as example. Assume that the actual expression of Eq. (2) is not known, but we somehow managed to obtain some data about it as in Table 3. The algorithm will firstly initialize the population randomly, decode individuals into expression trees and evaluate corresponding fitness according to the given data. For instance, a chromosome is initialized and decoded as in Fig. 3. Clearly, Fig. 3(b) can be further identified as Eq. (5) (Model A). Take a and b in Table 3
Table 2 Drilling data acquired from MD simulation. Statistical Parameter
Temperature (K)
Feed (m/s)
Rotational velocity (rad/s)
Mechanical strength (GPa)
Time (1012 s)
Mean Standard error Median Standard deviation Variance Kurtosis Skewness Minimum Maximum
370.00 6.41 375.00 43.01 1850.00 0.98 0.41 300.00 425.00
20.00 1.22 20.00 8.16 66.67 1.53 0 10.00 30.00
70.00 2.43 70.00 16.33 266.67 1.53 0 50.00 90.00
75.85 2.37 74.60 15.91 253.12 0.47 0.45 49.65 115.60
74.31 5.54 59.80 37.17 1381.44 1.19 0.62 31.64 147.50
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132
the training of models while the validation data is for testing and validating the models obtained. 4 Algorithm execution: The fundamental steps of the GEP algorithm can be represented as shown in Fig. 4. GEP starts with a random generated population with every individual representing a model for current dataset. Each model is then evaluated and ranked based on the fitness function. Roulette selection is employed to choose the individuals to be manipulated with genetic operators such as mutation, transposition, crossover/ recombination and so on, forming a new population of the same size. Mutation and crossover here are quite similar to those of GA's with the restriction that only terminal element can appear in the tail of a chromosome. Transposition operator randomly selects a sub-sequence of predefined length from a chromosome and moves it to the head. This loop of evolving new population continues until the stopping criterion is met. The best model of last generation is chosen as the final output of the algorithm.
Table 3 A simple dataset generated from Eq. (2). Input
Output
a
b
1 2 3
1 4 2
(a)
123
0 4 3
(b)
Fig. 3. A GEP chromosome of initialized population (a) Genotype (b) Phenotype.
as inputs and then we can obtain the output of this model/tree as shown in Table 4. If the sum of absolute error of the output is used as the fitness, comparing to the output of the dataset, we can easily calculate the fitness of this model is 14. After evaluation, the chromosomes will go through selection process and then be manipulated by operators to generate offspring. Again, consider the chromosome of Fig. 3, if the mutation operator is applied on the position 2 of the chromosome and randomly changes it from “b” to “-”, resulting in a different model (Model B). It can be found that though the encoding is slightly different from that of Fig. 2(a) in tails, but they do represent the same tree/model, i.e. Eq. (2), which is a perfect match for the dataset in Table 3.
f ða; bÞ ¼ a*b
(5)
3.1.3. Evolving models with Gene Expression Programming A general procedure of implementing GEP in modeling can be described as follows: 1 Definition of the function set and terminal set: The function set consists of arithmetic operators which may be used for modeling while the terminal set includes the independent variables of the dataset. Constants are also regarded as a terminal element which can be random numbers or a set of predefined values. 2 Determination of algorithm parameters and objective/fitness function: Usually, the parameters related to population and number of generation are set by user with the probabilities of evolutionary operators left by default. The objective function is then chosen to compute the fitness of obtained solutions and used to guide the evolutionary search towards the optimum. 3 Data partitioning: The dataset is usually divided into training set and validation set. The training data set is then used by GEP for
Fig. 4. Flowchart of gene expression programming.
Table 4 Output obtained by Model A and Model B. Target value
0 4 3
Model A
Model B
Output
Absolute error
Fitness
Output
Absolute error
Fitness
1 8 6
1 4 9
14
0 4 3
0 0 0
0
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3.2. Uncertainties in Gene Expression Programming The performance of GEP is influenced by the settings of its parameters. Different settings may result in a totally different result. Thus, the settings for GEP are regarded as uncertainties during the modeling process. In this study, three different settings are chosen to find out the influence on the performance of the formulated models, i.e. head size, number of generations and objective function. The objective function plays a vital role in guiding evolution of the population since the fitness is assigned by objective function and individuals with higher fitness are more likely to survive. Therefore, the fitness function greatly impacts the performance of GEP. Usually, many iterations may improve the results but requires more computation resources which influences the effectiveness and efficiency of the modeling process. Consequently, a proper value should be defined to guarantee the convergence while the computation time is constrained. Head size is a unique parameter in GEP which determines the length of the chromosome and a suitable value is needed to ensure that the optimal solution can be encoded within given length while maintaining the proper redundancy for individuals to evolve efficiently. In the following section, comparisons considering the mentioned uncertainties are performed.
3.3. Experimental settings and uncertainties being considered The overall experimental settings are listed as Table 5. Three sets of experiments are designed to investigate how these uncertainties influence the performance of GEP. For each experiment, a total number of 30 independent runs are chosen and the corresponding results are recorded. The statistical results namely average coefficient of determination (R2 ) is used for performance evaluation of the models.
3.3.1. Objective functions Firstly, different objective functions are set for each experiment while keeping other parameters fixed at values as described in Table 5 with number of generations set at 70. The reason for choosing smaller value for iterations is to find out which objective function converges fastest so that the process of modeling can be enhanced. Definitions of considered objective functions are given in Table 6, where RMSEps is a counterpart of RMSE with parsimony pressure. The datasets are partitioned into training set and testing set randomly as mentioned in Section 2. Statistic results of 30 runs are shown as Table 7 with bold numbers highlighting the best R2 obtained for the corresponding case. It is found that both RMSE with or without parsimony pressure outperforms the other two objective functions. Though RMSE and RMSEps have similar performance with RMSE slightly better in terms of accuracy, RMSEps takes program size into consideration and thus can generate formulas with simpler representations. Therefore, RMSEps is chosen as fitness function for further analysis.
3.3.2. Number of generations In context of generations, similarly, parameters are kept the same with RMSEps as fitness function while the only difference between runs is the number of generations. Multiple sets of generations are designed to find a suitable number of iterations for each case, namely to ensure an average R2 of 0.8. For the sake of clarity, only results of generations around best values are given in Table 8. It can be found that, basically, with the increment of number of generations, the R2 also increases which implies that better models are found. However, when the number of generations reaches a certain level, the rate of increase of R2 decreases. Thus, in this case, number of 200, 800 and 1200 generations are large enough for evolving models of high accuracy for the face milling energy, drilling operation strength and operation time
Table 5 Experimental settings for Gene Expression Programming. Function Set
Symbol
Terminal Set
Symbol
Addition Subtraction Multiplication Division Exponential Natural logarithm x to the power of 2
þ e * / Exp Ln
Face milling Cutting speed Depth of cut Feed rate
x1 x2 x3
Sine Cosine General Settings Population size Number of genes Constants per gene Operator Settings Mutation: Fixed-Root Mutation: Function Insertion: Leaf Mutation: Biased Leaf Mutation: Conservative Mutation: Conservative Fixed-Root Mutation: Conservative Function Mutation: Permutation: Conservative Permutation: Biased Mutation: IS Transposition: RIS Transposition: RNC Mutation: Constant Fine-Tuning: Constant Range Finding: Constant Insertion:
x2 Sin Cos
Drilling Temperature
x1
Feed Rotational velocity
x2 x3
30 3 10
Head size Linking Function Constants interval
7 þ [10,10]
0.00138 0.00068 0.00206 0.00546 0.00546 0.00364 0.00182 0.00546 0.00546 0.00546 0.00546 0.00546 0.00546 0.00206 0.00206 0.00009 0.00123
Inversion: Tail Inversion: Tail Mutation: Stumbling Mutation: Uniform Recombination: Uniform Gene Recombination: One-Point Recombination: Two-Point Recombination: Gene Recombination: Gene Transposition: Random Chromosomes: Random Cloning: Best Cloning: Dc Mutation: Dc Inversion: Dc IS Transposition: Dc Permutation:
0.00546 0.00546 0.00546 0.00141 0.00755 0.00755 0.00277 0.00277 0.00277 0.00277 0.0026 0.00102 0.0026 0.00206 0.00546 0.00546 0.00546
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132 Table 6 Objective/Fitness functions being considered. Objective Name (E)
Table 9 Performance comparison of different head size based on R2 .
Definition
Head size
qffiffiffiffiffiffi
RMSE
SSE N
PRESS FPE Fitness Function RMSEps
SSE N
1 þ 2k N
SSE N
Nþk Nk
5 6 7 8 9 10 11
1 $ 1 þ 1 $ kmax k 1000$1þE , 5000 k k max
125
Face milling
Drilling strength
Drilling time
training
testing
training
testing
training
testing
0.789599 0.807719 0.821557 0.816464 0.809168 0.802066
0.911788 0.912991 0.919698 0.913779 0.914333 0.907763
0.654676 0.802785 0.819449 0.784470 0.728971 0.746757
0.602094 0.767343 0.735656 0.680072 0.566847 0.641155
0.759214 0.822761 0.840658 0.807652 0.795476 0.809062
0.722875 0.820694 0.779031 0.789987 0.786481 0.755773
min
where kmax ¼ GðH þ tÞ; kmin ¼ G Others
1 1000$1þE
Note: SSE is sum of squares of error; k is program size of GEP tree; N is number of training samples; kmax and kmin represent maximum and minimum program sizes respectively; G is number of genes.
corresponding to R2 marked in bold are chosen for each case in the following modeling process to achieve the best performance. Finally, through the mentioned experiments, uncertainties in the GEP modeling process are explored. The tuned parameters for each case is summarized as Table 10.
respectively. 4. Simulation design and analysis 3.3.3. Head size It has been found that the head size of chromosome in GEP has a great impact on the algorithm where a too small or too large value may result in poor performance of models. Therefore, the head size is considered as another important uncertainty that should be taken good care of. With parameters set based on the findings of the two prior experiments and Table 5, different head sizes are used to explore the suitable values for each case. Statistical results as shown in Table 9 further validates the previous studies in literature that the relationship between head size and algorithm performance shows a similar trend to a concave upward function (Ferreira, 2006). Especially for the drilling strength case, it is observed that a small head size of 5 causes a significant drop in R2 for both training set and validation set, which implies that the chromosome is too short to encode all the information needed to fully describe the target model. Accordingly, R2 gradually drops if the head size is too large, which brings too much redundancy and causes expansion of search space. Thus, the head sizes
Table 7 Performance comparison of different objective functions based on R2 . Data Set
RMSE RMSEps PRESS FPE
Face milling
Drilling strength
Drilling time
training
testing
training
testing
training
testing
0.754205 0.744335 0.667615 0.603861
0.876609 0.868051 0.808952 0.672498
0.274167 0.243234 0.133083 0.107556
0.218560 0.215868 0.271440 0.294212
0.368709 0.332713 0.115668 0.026080
0.376706 0.360360 0.099850 0.039329
4.1. Model generation and selection In the previous section, the uncertainties of GEP have been explored and an additional 100 runs are performed to acquire the five best models for each case. To validate the obtained models, namely to check if the outputs are within the sensible interval ranges, 8000 inputs samples are generated by assuming it a normal distribution. The statistical results of each model in 3000 validation runs are listed as shown in Table 11. Though, Model 1 obtained from face milling dataset claims the highest R2 , negative values are found in this model which is meaningless since energy consumption should always be positive. With respect to the other four models, no negative output is found and the frequency histograms of these models are illustrated in Fig. 5. Fig. 5 summarizes output values from the GEP models so that their distributions can be easily observed. Since the input values are sampled from normal distributions within certain intervals, it can be expected that the output values of models are more sensible to share similar trends as the inputs. Thus, it can be observed again that a higher R2 does not always imply a better model due to a stronger normal trend is found in Model 5, which has a relatively smaller R2. For the other three models, the outputs are quite centralized compared to Model 5. Therefore, Model 5 is chosen as the final model for energy consumption of face milling process. Similarly, Model 3 and Model 2, which possess good balance between invalid outputs and normal trend, are picked as the best models for drilling mechanical strength and time respectively. Details of chosen models can be found in Appendices. The reason for existence of invalid outputs
Table 8 Performance comparison of different generations based on R2 . Number of Generations
100 200 500 600 700 800 900 1000 1100 1200 1300
Face milling
Drilling time
Drilling strength
training
testing
0.770230 0.820411 0.834009
0.881639 0.930355 0.935569
training
testing
0.639269 0.737960 0.764882 0.815429 0.837152
0.511783 0.720173 0.639295 0.773609 0.815841
training
testing
0.751204 0.765594 0.783571 0.824006 0.847916
0.725675 0.754938 0.789772 0.814897 0.865995
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Table 10 Fine tuned settings for face milling and drilling operation modeling. Data Set
Objective Function
Maximum iterations
Head size
Energy consumption in face milling Drilling strength Drilling time
RMSEps RMSEps RMSEps
500 800 1200
8 7 8
Table 11 Validation check of best five GEP models from 3000 runs. Data Set
Percentage of invalid outputs and training R2 Model 1
Energy consumption in face milling Drilling strength Drilling time
Model 2 (R2 ¼0.890)
2.8025% 0.8273% (R2 ¼0.959) 0.149% (R2 ¼0.964)
(R2 ¼0.870)
0% 0.0539% (R2 ¼0.942) 3.3824% (R2 ¼0.931)
Model 3
Model 4
(R2 ¼0.868)
0% 0.1381% (R2 ¼0.941) 27.994% (R2 ¼0.946)
(R2 ¼0.867)
0% 0.6154% (R2 ¼0.936) 3.081% (R2 ¼0.935)
(a) Model 1
(b) Model 2
(c) Model 3
(d) Model 4
(e) Model 5 Fig. 5. Output distribution of GEP models for energy consumption in face milling.
Model 5 0% (R2 ¼0.867) 0.2959% (R2 ¼0.935) 0.012% (R2 ¼0.931)
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132 Table 12 Prediction values obtained from the best model for face milling energy consumption.
127
Table 14 Prediction values obtained from the best model for drilling time.
Test no.
Target
Prediction
Test no.
Target
Prediction
Test no.
Target
Prediction
Test no.
Target
Prediction
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
311 294 321 274 371 431 393 412 498 482 458 521 754 632 547 839 440 406 365 326 453 406 501 549 535 680 486 625 851 653 773 582
341.93 311.45 372.63 281.21 348.92 493.03 396.67 444.71 558.67 489.34 420.35 628.32 685.22 590.61 496.37 780.21 415.29 385.70 356.33 327.20 449.80 403.74 496.13 542.75 550.48 684.65 483.89 617.40 842.77 659.62 751.01 568.61
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
499 368 431 473 609 456 514 664 620 503 668 516 693 625 785 876 534 403 463 502 652 724 524 590 547 748 650 782 632 893 693 757
458.49 373.33 401.48 429.87 548.09 458.74 503.27 593.18 612.06 741.88 676.81 547.65 729.19 641.13 817.62 906.44 502.29 419.60 446.93 474.50 600.60 644.36 513.92 557.12 611.64 736.90 674.10 800.02 713.93 971.22 799.32 885.08
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
147.50 140.30 135.80 73.80 70.20 65.90 49.20 42.50 37.10 137.11 129.15 123.50 68.39 63.21 59.80 45.93 39.21 35.32 130.65 122.14 116.26 65.17 59.67 56.10 43.69 37.34 34.02 124.50 115.20 108.90 62.30 55.60
126.70 129.98 133.27 76.41 67.40 58.39 59.52 46.76 34.00 117.17 120.46 123.74 71.52 62.50 53.49 56.17 43.41 30.65 111.68 114.96 118.24 68.70 59.69 50.68 54.26 41.49 28.73 105.85 109.13 112.42 65.73 56.72
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
52.50 41.50 35.40 33.10 116.49 107.57 100.75 58.14 52.18 47.19 38.27 33.76 31.64 102.80 95.60 84.30 51.40 45.30 40.80 34.70 45.90 24.60 62.31 50.96 41.73 28.92 24.20 12.46 22.45 16.94 14.13
47.71 52.23 39.47 26.71 99.84 103.12 106.40 62.67 53.66 44.65 50.16 37.40 24.64 82.31 85.59 88.88 53.78 44.77 35.76 44.15 31.39 18.63 66.45 69.73 73.01 45.79 36.78 27.77 38.78 26.02 13.26
Table 13 Prediction values obtained from the best model for drilling mechanical strength. Test no.
Target
Prediction
Test no.
Target
Prediction
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
115.60 102.50 93.30 103.80 92.40 84.90 103.20 92.70 84.50 103.14 85.35 71.12 87.03 76.19 67.98 87.38 77.56 66.22 92.83 77.17 65.03 79.58 71.39 61.28 82.02 69.61 59.20 88.30 71.90 59.10 74.60 64.90
105.91 92.04 86.31 97.97 87.00 82.83 92.93 83.51 80.27 99.41 84.42 77.87 90.15 78.54 73.81 84.27 74.47 70.83 94.99 79.47 72.52 85.06 73.17 68.16 78.76 68.81 64.97 90.19 74.16 66.81 79.60 67.44
33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
55.70 74.60 64.90 55.70 81.36 65.24 53.51 59.82 60.78 50.94 70.87 58.32 49.65 64.70 55.40 43.80 51.60 47.10 40.40 37.90 29.50 20.80 52.15 40.17 24.88 42.16 34.81 20.13 39.18 29.81 18.02
62.16 72.88 62.79 58.75 85.22 68.70 60.94 73.97 61.56 56.01 66.83 56.62 52.39 70.75 52.78 43.83 57.52 44.38 38.02 49.11 38.57 33.77 55.20 35.34 24.79 39.32 25.25 17.82 29.24 18.28 12.71
may rely on inefficient training of GEP models to encompass entire domain of the system. However, considering only limited input
values used for training, we can still conclude that the proposed approach performs reasonable well for both the case studies. Finally, comparisons between actual/target values and prediction values obtained by above chosen best models for each case study are shown in Table 12, Tables 13 and 14 respectively. 4.2. Sensitivity analysis To further explore the fundamental effects of each input element, multiple curves (Fig. 6) representing how these elements interact with outputs are plotted by fixing other inputs at mean values. For the face milling process, it can be observed in Fig. 6 (a) that all three inputs, namely depth of cut, feed rate and cutting speed, have considerable influence on energy consumption in a linear monotone increasing manner. Meanwhile, a higher variation in drilling mechanical strength is found with an increase in value of rotational velocity (Fig. 6 (b)), which indicates that rotational velocity is highly sensitive when compared to other two inputs. Fig. 6 (b) also suggests that the mechanical strength decreases nonlinearly with an increase in value of feed and temperature. With respect to drilling time, the relationship between each input factor is shown in Fig. 6 (c). In this case, the temperature becomes the major input element in drilling time model, leading to a higher fluctuation in operation time. Fig. 6 (c) also implies drilling time drops non-linearly with increment in feed and rotational velocity. From the above observation, the operation time can be shortened with trade-off in mechanical strength, states that the efficiency of process usually conflicts with product quality and thus further validates the rationality of GEP models. Similarly, 3D surface plots are used to illustrate the interaction effect of different inputs by fixing one of the inputs at its mean value. It can be observed in Fig. 7 (a) that each individual input has similar effect on energy consumption with combination of feed rate
(a) Energy consumption
(b) Mechanical strength of drilling process
(c) Operation time of drilling process Fig. 6. 2D plots illustrating relationship between inputs and outputs of different models.
(a) Energy consumption in face milling
(b) Mechanical strength of drilling process
(c) Operation time of drilling process Fig. 7. 3D plots illustrating relationship between inputs and outputs of different models.
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129
study proposes an evolutionary approach in design of robust energy consumption models considering uncertainties in the experiment and modeling procedures. The main findings from this work are as follows.
Fig. 8. Pareto front found by NSGA-II for drilling models.
and cutting as the dominant inputs. With respect to drilling operation, the combined effect of the rotational velocity and temperature generates higher non-linearity in mechanical strength of drilling process than the combination of other operating parameters. However, the combined effect of the feed and temperature generates higher non-linearity than the combination of other operating parameters in case of operation time. In order to obtain the optimum machining parameters, genetic algorithm is generally employed. However, for energy consumption in face milling, the optimum inputs can be determined by simply setting them as small as possible based on product requirements due to its linearity. The drilling case here is treated as a bi-objective optimization problem and thus NSGA-II, a popular multi-objective metaheuristic proposed by Deb et al. (2002), is used to find the Pareto front for drilling mechanical strength and time. Pareto front is a set of non-dominated solutions, being considered as optimal if no objective can be improved without trade-off at least one objective. Each solution in Pareto front weighs different objectives. In this case, for the drilling operation, decisions can be made by determining a lower boundary for the mechanical strength so that the quality of work piece can be guaranteed and then we may choose the non-dominated solution with minimum processing time and satisfactory mechanical strength. For example, if a minimum strength of 101 GPa is required, then the solution marked with red circle is chosen as shown in Fig. 8. Thus, the operating temperature of 304.72 K, feed rate of 16.87 m/s and rotational velocity of 54.89 rad/s are considered the optimum machining parameters to achieve a satisfactory mechanical strength of 101.08 GPa with a minimum processing time of 65:26 1012 s. In other words, maximum efficiency is achieved and corresponding energy utilization is optimized.
C In this study, prior experiments are carried out to deal with uncertainties of the algorithm, namely selection of objective function, number of iterations and head size determination. Experimental result shows that the objective function and head size have considerable influence on the performance of the algorithm for the given number of iterations. C Validation experiments imply that the GEP models are of good accuracy and performed reasonable well for the two case studies being considered. Models of good accuracy can be successfully obtained within pre-set relatively small iterations. C The behavior of obtained models are further explored using 2D and 3D plots, which reveal the sensitivities of models to different inputs so that the dominant input is determined. Through these graphs, the fundamental relationship between all the inputs can be clearly seen and thus can be used as guidance for actual productions and model validation. Furthermore, metaheuristics are employed to search for optimum parameters for the model in achieving lower energy consumption for both case studies. C The models formulated (Appendices) are explicit and can be coded into the system for online monitoring. The models can also be used in shop-floor of the production industry by workers/managers. Meanwhile, the present study possesses some limitations which can be treated as future research directions. To further improve the training of GEP models in case of limited data samples, Monte Carlo sampling technique can be integrated in framework of GEP to generate the moderate size data. Thus, the robustness and accuracy of model can be further enhanced.
Acknowledgments This research work was supported by National Natural Science Foundation of China (NSFC) under Grant nos. 51435009, Foundation for Innovative Research Groups of the National Natural Science Foundation of China under Grant nos. 51421062 and Shantou University Scientific Research Foundation (NTF 16002).
Appendices 5. Conclusions Appendix A Sustainable manufacturing plays a vital role in production industry due to the prominent problem of environment. The present
Table A.1 Dataset obtained from face milling experiments. Test no.
V(m/min)
ap (mm)
ft (mm/t)
EC(w)
Test no.
V(m/min)
ap (mm)
ft (mm/t)
EC(w)
1 2 3 4 5 6 7
157 157 157 157 188.4 188.4 188.4
0.6 0.6 0.6 0.6 0.6 0.6 0.6
0.15 0.1 0.2 0.05 0.05 0.2 0.1
311 294 321 274 371 431 393
33 34 35 36 37 38 39
157 157 157 157 188.4 188.4 188.4
1.2 1.2 1.2 1.2 1.2 1.2 1.2
0.2 0.05 0.1 0.15 0.15 0.05 0.1
499 368 431 473 609 456 514
(continued on next page)
130
W. Liao et al. / Journal of Cleaner Production 172 (2018) 119e132
Table A.1 (continued ) Test no.
V(m/min)
ap (mm)
ft (mm/t)
EC(w)
Test no.
V(m/min)
ap (mm)
ft (mm/t)
EC(w)
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32
188.4 219.8 219.8 219.8 219.8 251.57 251.57 251.57 251.57 157 157 157 157 188.4 188.4 188.4 188.4 219.8 219.8 219.8 219.8 251.57 251.57 251.57 251.57
0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.6 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9 0.9
0.15 0.15 0.1 0.05 0.2 0.15 0.1 0.05 0.2 0.2 0.15 0.1 0.05 0.1 0.05 0.15 0.2 0.1 0.2 0.05 0.15 0.2 0.1 0.15 0.05
412 498 482 458 521 754 632 547 839 440 406 365 326 453 406 501 549 535 680 486 625 851 653 773 582
40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64
188.4 219.8 219.8 219.8 219.8 251.57 251.57 251.57 251.57 157 157 157 157 188.4 188.4 188.4 188.4 219.8 219.8 219.8 219.8 251.57 251.57 251.57 251.57
1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.2 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5 1.5
0.2 0.1 0.2 0.15 0.05 0.1 0.05 0.15 0.2 0.2 0.05 0.1 0.15 0.15 0.2 0.05 0.1 0.05 0.15 0.1 0.2 0.05 0.2 0.1 0.15
664 620 503 668 516 693 625 785 876 534 403 463 502 652 724 524 590 547 748 650 782 632 893 693 757
Note: V represents cutting speed, ap is depth of cut, ft is feed rate and EC stands for energy consumption.
Appendix B
Table B.1 Dataset obtained from face milling experiments. Test no
Temperature (K)
Feed (m/s)
Rotation Speed (rad/s)
Strength (Gpa)
Time (1012 s)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
300 300 300 300 300 300 300 300 300 350 350 350 350 350 350 350 350 350 375 375 375 375 375 375 375 375 375 400 400 400 400 400 400 400 400 400 425
10 10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30 10
50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50
115.6 102.5 93.3 103.8 92.4 84.9 103.2 92.7 84.5 103.14 85.35 71.12 87.03 76.19 67.98 87.38 77.56 66.22 92.83 77.17 65.03 79.58 71.39 61.28 82.02 69.61 59.2 88.3 71.9 59.1 74.6 64.9 55.7 74.6 64.9 55.7 81.36
147.5 140.3 135.8 73.8 70.2 65.9 49.2 42.5 37.1 137.11 129.15 123.5 68.39 63.21 59.8 45.93 39.21 35.32 130.65 122.14 116.26 65.17 59.67 56.1 43.69 37.34 34.02 124.5 115.2 108.9 62.3 55.6 52.5 41.5 35.4 33.1 116.49
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Table B.1 (continued ) Test no
Temperature (K)
Feed (m/s)
Rotation Speed (rad/s)
Strength (Gpa)
Time (1012 s)
38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63
425 425 425 425 425 425 425 425 500 500 500 500 500 500 500 500 500 600 600 600 600 600 600 600 600 600
10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30 10 10 10 20 20 20 30 30 30
70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90 50 70 90
65.24 53.51 59.82 60.78 50.94 70.87 58.32 49.65 64.7 55.4 43.8 51.6 47.1 40.4 37.9 29.5 20.8 52.15 40.17 24.88 42.16 34.81 20.13 39.18 29.81 18.02
107.57 100.75 58.14 52.18 47.19 38.27 33.76 31.64 102.8 95.6 84.3 51.4 45.3 40.8 34.7 45.9 24.6 62.31 50.96 41.73 28.92 24.2 12.46 22.45 16.94 14.13
Appendix C
Energy consumption ¼ ðdð1Þdð2ÞÞ þ ððððdð1Þ þ dð2ÞÞ=ðG2C5 dð2ÞÞÞ ððG2C2G2C9Þ þ ðdð3Þ þ G2C5ÞÞÞððG2C5
G2C7 ¼ 5.67190810876797, G2C6 ¼ 7.09601123081149, G2C1 ¼ 0.408642841883602, G3C2 ¼ 0.777916806543168, G3C4 ¼ 5.27279915768914, G3C0 ¼ 0.573442793053987, d(1) is temperature, d(2) is feed and d(3) is rotational velocity.
þ dð1ÞÞðdð3Þ=G2C9ÞÞÞ þ ððððdð1Þ þ expðdð2ÞÞÞðlnðdð1ÞÞdð3ÞÞÞ þ expð
Appendix E
ðdð2Þ þ G3C1ÞÞÞ þ dð1ÞÞ (C.1)
where.
G2C5 ¼ 8.26105533005768, G2C9 ¼ 3.21284127323222, G2C2 ¼ 4.34369945371868, G3C1 ¼ 8.28012939848018, d(1) is cutting speed, d(2) is depth of cut and d(3) is feed rate.
Drilling time ¼ ðððdð2ÞG1C1Þ ðG1C2G1C8ÞÞ ððdð3Þ ^ G1C8Þ cosðdð1ÞÞÞÞ þ exp G2C42 . þ ððdð3Þ G2C3ÞG2C5Þ dð2Þ þ ð ðcosðdð1ÞÞðdð1Þ=dð2ÞÞÞ þ ðdð3Þ=ðG3C3 þ dð2ÞÞÞÞ where.
(E.1)
Appendix D
Drilling strength ¼
G1C1 ¼ 0.148716696676534, G1C2 ¼ 9.87925016022217, G1C8 ¼ 9.44456312753685, G2C5 ¼ 9.66063417462691, G2C4 ¼ 2.37126682332835, G2C3 ¼ 9.4550371929075, G3C3 ¼ 4.95236249732963, d(1) is temperature, d(2) is feed and d(3) is rotational velocity.
^ sin ln dð1Þ2 ðsinðdð3ÞÞ ðG1C3G1C3ÞÞ þ ðððG2C7dð1ÞÞ =cosðG2C6ÞÞ=ððdð2Þ þ dð3ÞÞ þ ðdð2Þ þ G2C1ÞÞÞ þ ð ðsinðdð3ÞÞsinðG3C2ÞÞðsinðdð1ÞÞ
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þ ðG3C4 þ G3C0ÞÞÞ where. G1C3 ¼ 8.66756187627796,
(D.1)
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