Sequencing the features to minimise the non-cutting energy consumption in machining considering the change of spindle rotation speed

A considerable amount of energy consumed by machine tools is attributable to non-cutting operations, including tool path, tool change, and change of spindle rotation speed. The non-cutting energy consumption of the machine tool (NCE) is affected by the processing sequence of the features of a specific part (PFS) because the plans of non-cutting operations will vary based on the different PFS. This article aims to understand the NCE between processing a specific feature and its pre- or post-feature, especially the energy consumed during the speed change of the spindle rotation. Based on the developed model, a single objective optimisation problem is introduced that minimises the NCE. Then, Ant Colony Optimisation (ACO) is employed to search for the optimal PFS. A case study is developed to validate the effectiveness of the proposed approach. Two parts with 12 and 15 features are processed on a machining centre. The simulation experiment results show that the optimal or near-optimal PFS can be found. Consequently, 8.70% and 30.42% reductions in NCE are achieved for part A and part B, respectively. Further, the performance of ACO for our specific optimisation problem is discussed and validated based on comparisons with other algorithms.


Introduction
According to the International Energy Agency, nearly 1/3 of the global energy consumption and 36% of carbon dioxide emissions are attributable to manufacturing industries [1], and the electricity consumption of machine tools accounts for more than half of the total U.S manufacturing electricity consumption [2]. Thus, reducing the energy consumption of machine tools during the use phase is a significant topic for both academic research [3] and industrial application [4]. The energy consumption of machine tools can be reduced by replacing the existing traditional machines with the advanced energy-efficient machines that have the energy-recycling devices [5] and the efficient power generation [6] and distribution [7] systems, but it greatly increases the financial burden on the enterprises and it is not economically sound for the enterprises to abandon the existing machines [8]. Considering economic efficiency, our research aims to reduce the energy consumption of the existing machine tools without purchasing additional energysaving devices.
The machining energy consumption of a machine tool (MTE) accounts for a majority of its total energy consumption [2]. The MTE is defined as the energy consumed by the machine tool for completing a feasible processing plan for a specific part, which can be divided to two types: non-cutting energy consumption and cutting energy consumption of the machine tool (NCE and CE) [9]. The NCE is defined as the energy consumed for the non-cutting operations of the machine tool, including tool path, tool change, and change of spindle rotation speed [10]. Generally, the NCE accounts for more than 30% of the MTE [10]. The CE is defined as the energy consumed when a part is actually cut by a machine tool. It has been proved that the value of the CE is affected by the processing sequence of the features of a part (PFS) [11]. Thus, finding the PFS which results in a smaller value of CE has been confirmed to be an effective energy consumption reduction approach [12]. However, the potentiality for this approach to reduce the NCE has not been well explored. Hu et al. [13] considered both the NCE and the CE while adjusting the PFS to reduce the MTE, but the detailed model for the NCE has not been provided. Besides, the CE model is redundant for the part without volumetric interaction among the features, but it has not been identified and removed from the existing MTE model.
For the NCE, the modelling work for the energy consumption of the machine tool during tool path (TPE) and tool change (TCE) has been developed by Hu et al. [14]. The TPE is defined as the energy consumed by the machine tool for moving the cutter to the right position to begin the actual cutting and the TCE is defined as the energy consumed by the machine tool for changing and selecting the right cutter [14]. However, the energy consumption of a machine tool during the change of spindle rotation speed (SCE) has been ignored. The SCE is defined as the energy consumed by the machine tool when the spindle rotates from a low (high) speed to a high (low) speed [15]. The SCE accounts for nearly 14% of the total NCE [10] and has energy-saving potentials [15]. The SCE can be subdivided into energy consumptions of the machine tool for the spindle acceleration (ASE) and deceleration (DSE). The PFS can affect the value of the SCE within the NCE, because the difference between the spindle rotation speeds of a pair of features on the PFS can vary if any of the features is replaced by another feature. Based on this discovery, the main novelty of this paper is to reduce the NCE with the SCE included through feature sequencing, and the proposed model and optimisation approach are the main contributions. The SCE can be directly obtained by the experiment measurements according to the start and end speeds. When using this method, the experiment measurements must be conducted again once the value of the start or end speed is changed, and it is laborious. To reduce the experiment costs, it is an innovation of this paper to introduce an empirical model to predict the SCE. It should be noted that the experiment is also required to develop the empirical model, but after obtaining the model, the SCE between any two spindle rotation speeds can be predicted without further experiments. In the optimisation work, Hu et al. [13] has verified that Genetic Algorithm (GA) can effectively solve the energy-aware feature sequencing problem when the MTE is regarded as the optimisation objective. When the optimisation objective is changed to the NCE which considers the SCE, the performance of GA may become inferior. A purpose of the optimisation work delivered in this paper is to present and validate an effective algorithm for solving the new single objective optimisation problem.
Based on the above, this study aims at understanding the SCE and integrating the developed SCE model with the existing NCE model which only considered the TPE and the TCE. Then, a model to depict the NCE between processing a specific feature and its pre-or post-feature has been further developed. The single objective optimisation in this research is to minimise the NCE through finding the optimal PFS. According to preliminary studies, Ant Colony Optimisation (ACO) is employed and modified as the optimisation approach to search for the optimal PFS for its good performance in solution quality [16] and computation speed [17]. Based on the case study, the proposed approach is demonstrated and its performance is compared and validated. In this study, it is assumed that all of the required processing for a part can be finished on a single machine tool.
The remainder of the paper is organised as follows. The background and motivation are given in the next section. The problem description and the model for the NCE are presented in Section 3. In Section 4, the working procedures of ACO for solving the aforementioned optimisation problem are described. Case studies are conducted to demonstrate and discuss the developed approach in Section 5, and a brief summary and future work are given in Section 6.

Background and motivation
The reduction of NCE has been the topic in the previous energyaware feature sequencing studies. For example, a mathematic model was developed to reduce the NCE, including the TPE and the TCE by adjusting the PFS [14]. However, the model ignored the SCE resulting from the difference between the spindle rotation speeds of a pair of features on the PFS. The SCE accounts for nearly 14% of the total NCE [10], and it has energy-saving potentials [15]. In related studies considering the SCE, a mathematic model was developed where the value of the SCE was assumed as a constant [18]. In addition, a feature precedence graph was generated to identify the manufacturing precedence constraints, and the value of the SCE was also assumed as a constant [19]. A main limitation of these models is that they use an inaccurate and oversimplified SCE model. For example, the value of the SCE between all pairs of features in a part has not been set to a variable that considers the actual values resulting from the required acceleration or deceleration. In fact, the SCE is dependent on the start and end speeds of the spindle rotation between the features. Moreover, the data for the SCE were made up, which weakens the accuracy of the model and skews the results.
The value of the SCE can be accurately obtained by the experiment measurements according to the start and end speeds [15]. When using this method, the experiment measurement must be conducted again once the value of the start or end speed is changed. To reduce the experiment costs, it is important to develop a SCE model to predict the value of the SCE according to the difference between two spindle rotation speeds. To obtain the SCE model, Shi et al. [20] developed a quadratic model to predict the energy consumption of a machine tool for the acceleration of the spindle rotation (ASE) from measured power data. However, the start speed of the spindle rotation can only be set to 0 rpm. To predict the ASE from an arbitrary low speed to a higher speed, a model based on the spindle torque was proposed [21]. The coefficients in the model were obtained by the experiments, and the prediction accuracy achieved 90% [21]. However, the model is unable to predict the energy consumption of a machine tool for the deceleration of the spindle rotation (DSE). A model for the DSE was developed by multiplying the torque with the angular velocity [22], but the parameters used in this model, such as friction torque, are difficult to acquire for a specific machine tool. The results of these previous studies do not yet provide a comprehensive SCE model but can be used as precursor to develop our model for predicting the SCE between processing a specific feature and its pre-or post-feature.
After developing the mathematic model, the algorithms can be employed to search for the optimal PFS which results in the minimisation of the NCE with the SCE included. Hu et al. [13] has proved that Genetic Algorithm (GA) can effectively solve the energy-aware feature sequencing problem when the MTE is regarded as the optimisation objective. When the optimisation objective is changed to the NCE, the performance of GA may become inferior. So far, the specific algorithms to minimise the NCE have received little attention. For the related time-aware feature sequencing problem, plenty of algorithms, such as deterministic algorithms and metaheuristics, have been employed in the literature. These works can be used as references for the algorithms selection to minimise our objective. Traditionally, the deterministic algorithms, such as dynamic programming [23] and branch-and-bound method [24], have been used to find the global optimal PFS which results in the minimisation of machining time [23]. However, they are only suitable for solving the small-to-medium sized problems because the computation time sharply increases when the target part has more than 20 features [25]. For example, when the number of the features in a turned part increased from 13 to 16, the computation time of a deterministic algorithm sharply increased from 29.95 s to 1464.7 s [25]. Comparatively, the computation time of a metaheuristic increased from 0.81 s to only 1.66 s [26]. Metaheuristics have become increasingly popular because they require much shorter computation time than that of deterministic algorithms for the large problems [26]. However, meta-heuristics do not guaranty finding the optimal solution and can get trapped by local optima. Hence, there is a need to select a suitable metaheuristic for a given problem. Hu et al. [16] verified that Ant Colony Optimisation (ACO) can effectively solve the time-aware feature sequencing problem. In this article, the performance of ACO on solving our specific energy-aware feature sequence problem is compared and validated in terms of the optimality of the solutions and computation time.
According to the literature from industry and academia as reviewed above, the modelling for the NCE, in particular the SCE, is still not sufficient. Moreover, the optimisation approach for reducing the NCE through adjusting the PFS in a single machine environment has also not been well explored yet. These important gaps motivated the research presented in this paper. The developed model for the SCE and the corresponding optimisation approach based on ACO are introduced in following sections.

Problem statement and modelling
Considering a part, all of its n features can be denoted as a finite set F C ¼ fF i g n i¼1 . When machining the part, the value of the NCE is also affected by the start and end positions of the tool. Thus, they are defined as two virtual features for the part, denoted by F 0 and F nþ1 , respectively. In this research background, there is a finite set of n þ 2 features F ¼ fF i g nþ1 i¼0 for an n-feature part. The F C is a subset of the FðF C 3FÞ.
In Fig. 1, a part with two features (F 1 and F 2 ) is used as an illustrative example to show that the different PFSs can result in different values of NCE. The spindle rotation speeds for processing F 1 and F 2 are 500 rpm and 800 rpm, respectively. The start and end positions of the tool, which are virtual features, are denoted as F 0 and F 3 , respectively. Two PFSs can be adopted: The tool paths of these two PFSs are labelled by blue solid lines and red dashed lines, respectively, in Fig. 1. The spindle acceleration and deceleration during machining are marked as " " and " ", respectively. The corresponding power profiles of the machine tool when processing the aforementioned two PFSs are shown in Fig. 2. The power profiles are developed based on the measured data and the prediction method by Jia [10] and Dahmus and Gutowski [27].
In Fig. 2, the areas filled with blue or red nets represent TPE. The areas filled with forward slashes and back slashes represent ASE and DSE, respectively, and the blank areas represent CE. Normally, tool changes are required during machining which results in TCE, although it is not the case in this example. This research focuses on the effect of the PFSs on the NCE which is the sum of all the TPE, TCE, and SCE. The NCE between finishing cutting the feature F p and the beginning of cutting its post-feature F q on the sequence can be expressed as: can be found in Hu et al. [14]. By executing the noncutting operations from F p to F q , there can be more than one change for the spindle rotation speed. Thus, a finite set of energy consumption for m changes of spindle rotation speed is employed to indicate the SCE between two features. E ðFp;FqÞ src can be expressed as the following: where C ðFp;FqÞ j is the energy consumption for the j-th change of the spindle rotation speed during the non-cutting operations from F p to F q . The effect of the different PFSs on the value of the NCE and the SCE can be seen by comparing the filled areas in Fig. 2(a) and (b). The goal of this research is to determine the optimal PFS for a part that minimises the total NCE. All of the positions of the features in a sequence can be denoted as a finite set S ¼ fS l g nþ2 l¼1 , because as the aforementioned definition, there are n þ 2 features for an n features part. S l indicates the feature at the l-th position of a sequence. For example, S l ¼ F p indicates the feature at the l-th position of a sequence is F p . For any part, the feature at the 1-st position and the n þ 2-th position is F 0 (S 1 ¼ F 0 ) and F nþ1 (S nþ2 ¼ F nþ1 ), respectively. Then, the objective function for minimising the NCE based on a specific PFS can be expressed as: where E N is the total NCE based on a specific PFS and E ðS l ;S lþ1 Þ non is the NCE between the feature at the l-th position and the feature at the l þ 1-th position of a sequence. Its value can be obtained according to Expression (1). Constraints of the model are developed according to the precedence constraints among the features [25]. A feasible PFS must satisfy all constraints. The total NCE for the corresponding PFS is set to infinity "∞" once any feature and its pre-or postfeatures in a sequence violate any constraint.
Then, in Expression (2), the C ðFp;FqÞ j can be expressed as [28]: where P pq Cj is the power of a machine tool during the j-th speed change of the spindle rotation in the non-cutting operations from F p to F q . The power of a machine tool during the speed change of the spindle rotation can be divided to two parts: the basic power of the machine tool and the power of the spindle system [27] as shown in Fig. 2. Thus, P pq Cj is expressed as: where P 0 is the basic power of the machine tool and P pq cj is the power of the spindle system during the j-th speed change of the spindle rotation in the non-cutting operations from F p to F q .
For a spindle acceleration, the model developed by Lv [21] can be employed and modified to model the P pq cj as: where n pq Sj is the initial speed of the spindle for the j-th speed change of spindle rotation, B SR and C SR are the monomial coefficient and constant in the model, which are obtained by linear regression based on the power data of machine tools [29], and a A and T s are the angular acceleration and the acceleration torque of a spindle, respectively, which are obtained by experiment measurements.
For a spindle deceleration, if there is no energy-recycling device installed on the machine tool, the power consumption during deceleration equals to the basic power of the machine tool. Thus, P pq cj ¼ 0. Otherwise, if the energy-recycling devices have been installed, the power level during deceleration can be negative as shown in Fig. 2 because the energy is recovered by the energyrecycling devices. Based on the measured power data, there is a linear relation between the average power of the spindle system and the speed interval of deceleration. Hence, a linear equation is employed to model P pq cj as: where n pq Ej is the end speed of the spindle for the j-th speed change of the spindle rotation; B SRD and C SRD are the monomial coefficient and constant, which are obtained by linear regression based on the measured power data.
In Expression (4), t pq Cj is the time for the j-th speed change of the spindle rotation during the non-cutting operations from F p to F q , which is calculated by: In the next section, the algorithm for finding the optimal PFS in terms of the NCE minimisation is introduced.

Solution algorithms
Ant Colony Optimisation (ACO) is selected to search for the optimal PFS, by using the minimisation of NCE as the objective. Besides, three other algorithms including Depth-First Search (DFS), Genetic Algorithm (GA) and Particle Swarm Optimisation (PSO) are used as the benchmarks for the comparison and verification of ACO. ACO is introduced in detail as follows.
ACO is a meta-heuristic and probabilistic optimisation technique that imitates the behaviour of ants to discover the best path between their colony and a source of food via artificial pheromone trails [30]. A flowchart of ACO is shown in Fig. 3. At the beginning, ACO parameters are set, including a, b, r, Q , and K to denote the relative importance of the pheromone, the relative importance of the heuristic information, the evaporation rate, the constant, and the number of ants, respectively. All of the K ants are placed in the starting feature and then each ant continues to select the next feature to be visited through a stochastic transition rule until all ants have reached the end feature. After all of the K ants have reached the end feature, the optimal paths till now are determined and the amount of pheromones on each edge between the features are updated according to a pheromone update rule. The process of all of the K ants moving from the starting feature to the end feature is regarded as an iteration. This iteration process is repeated until a stopping condition has been met. The stopping condition can be the specified maximum number of iterations reached.

The stochastic transition rule
Each ant selects the next feature to visit through a stochastic transition rule [17]. In this rule, the edge with more pheromones and heuristic information is more likely to be selected. Specifically, when the k-th ant is in the feature F p , the probability of going to feature F q is: where P k pq is the probability of going to the feature F q for the k-th ant in the feature F p , g is the index for a feature, N k p is the set of indices for the features not yet visited by the k-th ant, t pq is the amount of pheromones on the edge between F p and F q , which is updated according to Rule (10), h pq is the heuristic information on the edge between F p and F q , and a and b are the parameters to control the relative importance of the pheromone and the heuristic information, respectively. h pq is calculated by the reciprocal of

The pheromone update rule
At each iteration, the amount of pheromones on each edge between the features is updated according to the pheromone update rule [17], and the variables in the rule are obtained from the K paths (PFSs) constructed by the K ants in an iteration. The pheromone update rule is given by: Dt k pq (10) where r is the evaporation rate, K is the number of ants, and Dt k pq is the amount of pheromones laid on the edge (p, q) by the k-th ant at an iteration, which is calculated by: if the k À th ant used edgeðp; qÞin its path; 0 otherwises;  where Q is a constant and L k is the energy consumption for the k-th ant's path ðL k ¼ E N Þ.

Case description, modelling and optimisation
Two parts were used as the case studies to validate the developed mathematic model and the optimisation approach. Part A has 12 actual features (holes) denoted by F 1 eF 12 and 2 virtual features (F 0 and F 13 ), as shown in Fig. 4. Part B has 15 actual features denoted by F 1 eF 15 and 2 virtual features (F 0 and F 16 ), as shown in Fig. 5. For part B, the feature F 1 (plain) should be processed prior to any other features F i (2 i 15). A vertical machining centre (XHF-714F) manufactured by Hangzhou CNC Machine Tool Co., Ltd. of China was used to process the two parts. The experiment set-up for the power data collection on the XHF-714F is the same as that in Hu et al. [13]. The key parameters of the XHF-714F required for the calculation of the NCE are listed in Table 1. They have been obtained through experiment measurements and regression analysis based on the method developed by Lv [21]. The process parameters defining the spindle rotation speed for each feature in part A and part B are listed in Tables 2 and 3. They have been obtained from the process files. On the basis of the above and the additional case information provided in Hu et al. [14], the NCE can be calculated.  The spindle rotation speeds for F 1 and F 5 are 500 rpm and 700 rpm according to Table 2, so n 15 S1 ¼ 500 rpm and n 15 E1 ¼ 700 rpm and the 1-st change of the spindle rotation speed from F 1 to F 5 is acceleration. Then, Equations (4) and (5) where P 15 c1 is the power of the spindle system during the 1-st speed change of the spindle rotation in the non-cutting operations from F 1 to F 5 and t 15 C1 is the corresponding time for the speed change of the spindle rotation. According to Table 1, the basic power of the XHF-714F is P 0 ¼ 371.0W. According to Equation (6) Table 4. Moreover, the 256 NCE values for part B are directly given in Table 5.
The data in Tables 4 and 5 are the input data of the optimisation approach. Ant Colony Optimisation (ACO) is employed as our optimisation approach. ACO and all of the other algorithms are developed on a software platform Dev Cþþ 5.11.0 with the programming language Cþþ. The parameters of the computation facility used for the experiments are as follows: Intel (R) Core (TM) i7-2630 QM CPU with 2.00 GHz, 4.00 GB RAM, and Windows 7 (64bit). The parameter values used for the ACO are obtained by tuning, and their values are as follows: population size¼ 50, a ¼ 1:0, b ¼ 4:0, evaporation rate r ¼ 0:1, Q ¼ 500, and iter-ation¼ 300, as listed in Table 6. By running the developed ACO 20 times for part A and part B, the minimum NCE that ACO can  achieve is 49579J and 106703J, respectively. The corresponding PFS are F 0 eF 5 eF 4 eF 8 eF 9 eF 12 eF 11 eF 10 eF 3 eF 7 eF 2 eF 6 eF 1 eF 13 and F 0 eF 1 eF 2 eF 5 eF 6 eF 3 eF 4 eF 7 eF 10 eF 9 eF 8 eF 12 eF 13 eF 14 eF 15 e F 11 eF 16 , respectively. Besides, the average NCE for part A and part B is 49815J and 106703J, and the average computation time is 0.57s and 0.68s. The results are summarised in Table 7.

Results analysis and discussion
In this section, the potentiality and effectiveness of the proposed    approach in reducing the NCE is analysed and demonstrated. Besides, the performance of ACO for this specific optimisation problem is discussed and validated based on the comparison with other algorithms.

Energy savings benefit from our approach
To demonstrate the effectiveness of the developed approach in reducing the NCE, the following comparison is conducted. A PFS produced by the Bottom-to-Top (BTT) [31] serves as the benchmark to represent the traditional sequencing technique to arranging the PFS without the energy-saving consideration. The benchmark PFSs of part A and part B are F 0 eF 4 eF 12 eF 8 eF 5 eF 9 eF 3 eF 1 eF 10 eF 6 eF 7 eF 2 eF 11 eF 13 and F 0 eF 1 eF 2 eF 5 eF 12 eF 15 eF 10 eF 7 eF 3 eF 4 eF 11 eF 6 eF 14 eF 13 eF 8 eF 9 eF 16 , respectively. The NCE for the benchmark PFSs of part A and part B are 54300J and 153362J, respectively. In comparison, the minimum NCE for the PFSs of part A and part B based on our approach are 49579J and 106703J, respectively. Thus, 8.70% [(54300-49579)/54300] and 30.42% [(153362-106703)/ 153362] of the NCE for part A and part B can be saved. More percentage of the NCE for part B is saved than that of part A, partly because the difference between the spindle rotation speeds of the features in part B is larger than that of part A.
In addition, by using the sequencing approach developed by Hu et al. [13], the PFS of part B obtained is F 0 eF 1 eF 2 eF 11 eF 15 eF 14 eF 13 eF 12 eF 8 eF 9 eF 10 eF 7 eF 5 eF 6 eF 3 eF 4 -eF 16 and the NCE for this PFS is 108445J. Thus, compared with the approach developed by Hu et al. [13], our approach can save 1.61% [(108445-106703)/108445] more of the NCE for part B.

Verification of the performance of ACO
To verify the performance of ACO, a deterministic algorithm, Depth-First Search (DFS), is employed because it can always accurately find the global optimal solution [32]. Based on DFS, the global minimum NCE for part A and B are 49537J and 106703J, respectively.
The corresponding PFS for part A is F 0 eF 1 eF 6 eF 2 eF 7 eF 10 eF 11 eF 12 eF 9 eF 3 eF 8 eF 4 eF 5 eF 13 and the corresponding PFS for part B is the same as that produced by ACO. Based on the comparison with the results obtained using DFS, ACO only achieves the near-minimum NCE of 49579J for part A in 20 trials, and it is remarkable that ACO achieves the global minimum NCE of 106703J in each trial (20 times) for part B, as summarised in Table 7. Interestingly, the solutions of part B obtained using ACO are much better than that of part A, although the design of part B is more complex than that of part A. There is probably a relationship between the design of the parts and the performance of ACO in solution quality.
In average, the solutions obtained using ACO are only 0.562% [(49815-49537)/49537] and 0% [(106703-106703)/106703] inferior than the global optimum for part A and part B. Thus, the performance of ACO is excellent in solution quality for this specific Table 1 Parameters of the machining centre (XHF-714F) in the power models.   Table 3 Spindle rotation speed for each feature in part B.

Item
The i-th feature Spindle rotation speed [rpm] Table 4 Non-cutting energy consumption between the features in part A.

Comparison of ACO with other meta-heuristics
The performance of ACO is compared with other metaheuristics, such as the standard Genetic Algorithm (GA) and Particle Swarm Optimisation (PSO). It has already been verified by a previous study [13] that GA can effectively solve the energy-aware feature sequencing problem when MTE is regarded as the optimisation objective, thus GA is compared with ACO for solving the new single objective optimisation problem. Besides, PSO is selected and compared with ACO for its good performance in solving the timeaware feature sequencing problem [33]. The parameter values used for GA and PSO have been obtained by tuning and are listed in Table 6. Each algorithm runs 20 times for both part A and part B. The optimisation results using the three different meta-heuristics are summarised and compared in Table 7.
According to Table 7, ACO and GA consistently outperform PSO in all of the experiments for the two parts. Although GA performs better than ACO in solution quality for part A, its computation time is longer than that of ACO. For part B, ACO outperforms GA in both solution quality and computation time. Comparatively, ACO performs best among the three standard meta-heuristics. It should be noted that the meta-heuristics are compared under the condition of our specific algorithm parameters and experiments. Although some algorithm parameters have been tuned, the optimality of these parameters is not guaranteed in this article. GA and PSO may outperform ACO through modifying the algorithm parameters, such as the crossover and mutation operators, the particle update rules, and the corresponding values of probabilities [34], but these modifications are out of the scope of this article. Besides, some novel and advanced meta-heuristics, such as Hybrid Genetic Algorithm-Ant Colony Optimisation [35], Ant Lion Optimisation Algorithm [36], and Bird-Mating Optimisation [37], have not been employed and compared in this article. In the future, research work on exploring a better algorithm for solving our problem can be conducted.

Conclusions and future work
It has been confirmed that the machining energy consumption of the machine tool can be reduced by adjusting the processing sequence of the features of a part (PFS) at the process planning stage [13]. However, the NCE portion has not been well explored in the previous research. In particular, the effect of the PFS on the SCE has not been understood, and the SCE normally accounts for nearly 14% of the total NCE. Thus, this research develops a novel SCE model and integrates it with the existing TPE and TCE models [14] to obtain the completed NCE model. Based on this NCE model, a new single objective optimisation problem that minimises the NCE is introduced. Then, Ant Colony Optimisation (ACO) is employed and modified as the optimisation approach to search for the optimal PFS, and the performance of ACO is compared and validated. In summary, it is the main innovation of this paper to reduce the NCE with the SCE included through feature sequencing, and the proposed model and optimisation approach for the new problem are the main contributions.
In the case study, the optimal and near-optimal PFSs for two parts with 12 and 15 features have been found. Consequently, 8.70% and 30.42% of the NCE for part A and part B are reduced, which validates the effectiveness of the developed approach. Although the solutions obtained using ACO are 0.562% and 0% inferior than the global optimum for part A and part B, the computation time of ACO Table 5 Non-cutting energy consumption between the features in part B. In this presented research, it is laborious to calculate ðn þ 1Þ 2 NCE values one by one for a part with n actual features. Thus, the automation for the corresponding calculation can be improved. One limitation is that some other non-cutting operations, such as setup change, have not been considered. Usually, machine tools consume energy during the execution of these operations. Thus, for the next step, the energy consumption model for these operations will be developed. The single objective is another limitation. In real manufacturing circumstances, it is unrealistic to only reduce the NCE without controlling the processing time, quality, and cost. Thus, other optimisation objectives, including time, quality, and cost, should also be considered when optimising the NCE. In the future, the relationship between the design of the parts and the performance of ACO in solution quality will be explored. Besides that, research work on exploring a better algorithm for solving our problem will be conducted. Finally, the proposed energy-aware feature sequencing approach will be combined with the product design software to assist in industrial applications.
The abbreviations and notations used in the problem statement, the algorithm description and throughout the paper are as follows: index for a feature in a part F C a finite set of n features for a part, F C ¼ fF i g n i¼1 F 0 a virtual feature for a specific part to denote the start position of the tool F nþ1 a virtual feature for a specific part to denote the end position of the tool F a finite set of n þ 2 features for a part in the machining

Abbreviations
index for a change of the spindle rotation speed during the non-cutting operations from a feature to its postfeature m number of changes for the spindle rotation speed during the non-cutting operations from a feature to its postfeature S a finite set of n þ 2 positions of features in a sequence, S ¼ fS l g nþ2 l¼1 S l l-th position of a sequence l index for a position in a sequence   Supplementary data related to this article can be found at https://www.researchgate.net/publication/319059278_ Sequencing_the_features_to_minimise_the_non-cutting_energy_ consumption_in_machining_considering_the_change_of_spindle_ rotation_speed.