- Research
- Open Access
A split-optimization approach for obtaining multiple solutions in single-objective process parameter optimization
- Manik Rajora^{1},
- Pan Zou^{2},
- Yao Guang Yang^{3},
- Zhi Wen Fan^{3},
- Hung Yi Chen^{3},
- Wen Chieh Wu^{3},
- Beizhi Li^{2} and
- Steven Y. Liang^{1, 2}Email author
- Received: 31 March 2016
- Accepted: 17 August 2016
- Published: 26 August 2016
Abstract
It can be observed from the experimental data of different processes that different process parameter combinations can lead to the same performance indicators, but during the optimization of process parameters, using current techniques, only one of these combinations can be found when a given objective function is specified. The combination of process parameters obtained after optimization may not always be applicable in actual production or may lead to undesired experimental conditions. In this paper, a split-optimization approach is proposed for obtaining multiple solutions in a single-objective process parameter optimization problem. This is accomplished by splitting the original search space into smaller sub-search spaces and using GA in each sub-search space to optimize the process parameters. Two different methods, i.e., cluster centers and hill and valley splitting strategy, were used to split the original search space, and their efficiency was measured against a method in which the original search space is split into equal smaller sub-search spaces. The proposed approach was used to obtain multiple optimal process parameter combinations for electrochemical micro-machining. The result obtained from the case study showed that the cluster centers and hill and valley splitting strategies were more efficient in splitting the original search space than the method in which the original search space is divided into smaller equal sub-search spaces.
Keywords
- Split-optimization
- Multiple solutions
- Process parameter optimization
- Genetic algorithm (GA)
- Electrochemical micro-machining (EMM)
Background
In today’s rapidly changing scenario in the manufacturing industries, optimization of process parameters is essential for a manufacturing unit to respond effectively to the severe competitiveness and increasing demand for quality products in the market (Cook et al. 2000). Previously, to obtain optimal combinations of input process parameters, engineers used a trial-and-error-based approach, which relied on engineers’ experience and intuition. However, the trial-and-error-based approach is expensive and time consuming; thus, it is not suitable for complex manufacturing processes (Chen et al. 2009). Thus, researchers have focused their attention on developing alternate methods to the trial-and-error-based approach that can help engineers obtain the combination of process parameters that will minimize or maximize the desired objective value for a given process. The methods for obtaining these combinations of process parameters can be split into 2 main categories: 1. forward mapping of process inputs to a performance indicator with backwards optimization and 2. reverse mapping between the performance indicators and the process inputs. In forward mappings, first, a model is created between the process inputs and the performance indicators using either physics-based models, regressions models, or intelligent techniques. Once a satisfactory model has been created, it is then utilized to obtain the combination of process parameters that will lead to a desired value of the output using optimization techniques such as the Genetic algorithm (GA), Simulated Annealing (SA), Particle Swarm Optimization (PSO), etc. The desired output can either be to a. minimize a given performance indicator or b. reach a desired level of a performance indicator.
Chen et al. (2009) utilized the back propagation neural network (BPNN) and GA to create a forward prediction model and optimize the process parameters of plastic injection molding. Ylidiz (2013) utilized a hybrid artificial bee colony-based approach for selecting the optimal process parameters for multi-pass turning that would minimize the machining cost. Senthilkumaar et al. (2012) used mathematical models and ANN to map the relationship between the process inputs and performance indicators for finish turning and facing of Inconel 718. GA was then used to find the optimal combination of process parameters, with the aim of minimizing surface roughness and flank wear. Pawar and Rao (2013) applied the teaching–learning-based optimization (TLBO) algorithm to optimize the process parameters of abrasive water jet machining, grinding, and milling. They created physics-based models between the input and output parameters of each process and then utilized TLBO to minimize the material removal rate in abrasive water jets, minimize production cost and maximize production rate with respect to grinding, and minimize the production time in milling. Fard et al. (2013) employed adaptive network-based fuzzy inference systems (ANFIS) to model the process of dry wire electrical discharge machining (WEDM). This model was then used to optimize, using artificial bee colony (ABC), the process inputs that would minimize surface roughness and maximize material removal rate. Teixidor et al. (2013) used particle swarm optimization (PSO) to obtain optimal process parameters that would minimize the depth error, width error, and surface roughness in the pulsed laser milling of micro-channels on AISI H13 tool steel. Katherasan et al. (2014) used ANN to model the process of flux cored arc welding (FCAW) and then utilized PSO to minimize bead width and reinforcement and maximize depth of penetration. Yusup et al. (2014) created a regression model for the process parameters and process indicators of an abrasive waterjet (AWJ) and then used ABC to minimize the surface roughness. Panda and Yadava (2012) used ANN to model the process of die sinking electrochemical spark machining (DS-ESM) and then used GA for multi-objective optimization of the material removal rate and average surface roughness. Maji and Pratihar (2010) combined ANFIS with GA to create forward and backward input–output relationships for the electrical discharge machining process (EDM). In their proposed methodology, GA was used to optimize the membership functions of the ANFIS, with the aim of minimizing the error between the predicted and actual outputs. Cus et al. (2006) developed an intelligent system for online monitoring and optimization of process parameters in the ball-end milling process. Their objective was to find the optimal set of process parameters, using GA to achieve the forces selected by the user. Raja et al. (2015) optimized the process parameters of electric discharge machining (EDM) using the firefly algorithm to obtain the desired surface roughness in the minimum possible machining time. Raja and Baskar (2012) used PSO to optimize the process parameters to achieve the desired surface roughness while minimizing machining time for face milling. Rao and Pawar (2009) developed mathematical models using response surface modeling (RSM) to correlate the process inputs and performance indicators of WEDM. They then used ABC to achieve the maximum machining speed that would give the desired value of the surface finish. Lee et al. (2007) modeled the process of high-speed finish milling using a 2 stage ANN and then used GA to maximize the surface finish while achieving the desired material removal rate. Teimouri and Baseri (2015) used a combination of fuzzy logic and the artificial bee colony algorithm to create a forward prediction model between input and output parameters for friction stir welding (FSW). This trained model was then utilized to find the optimal input parameters that would give the desired output value by minimizing the absolute error between the predicted and specified output using the imperialist competitive algorithm (ICA).
An ample amount of work has also been done to create a reverse mapping model between the process parameters and the performance indicators. Parappagoudar et al. (2008) utilized the back-propagation neural network (BPNN) and a genetic-neural network (GA-NN) for forward and reverse mapping of the process parameters and performance indicators in a green sand mold system. Parappagoudar et al. (2008) also extended their application of BPNN and GA-NN to create forward and backward mappings for the process of the Sodium Silicate-Bonded, Carbon Dioxide Gas Hardened Molding Sand System. Amarnath and Pratihar (2009) used radial basis function neural networks (RBFNNs) for forward and reverse input–output mapping of the tungsten inert gas (TIG) welding process. In their proposed methodology, the structure and the parameters of the RBFNN were modified using a combination of GA and the fuzzy C-means (FCM) algorithm for both the forward and reverse mapping. Chandrashekarappa et al. (2014) used BPNN and GA-NN for forward and reverse mappings of the squeeze casting process. Kittur and Parappagoudar (2012) utilized BPNN and GA-NN for forward and reverse mapping in the die casting process. Because batch training requires a tremendous amount of data, they used previously generated equations to supplement the experimental data. Malakooti and Raman (2000) used ANN to create forward- and backward-direction mappings between the process outputs and inputs for the cutting operation on a lathe.
Even though extensive research has been done regarding optimization of the process parameter for different processes, the current algorithms used for the optimization procedure are limited to finding only one set of optimal process parameter combinations for a single-objective optimization problem each time the algorithms are executed. Though this process parameter combination may achieve the desired output, it may not always be suitable for actual production or may lead to undesirable experimental conditions. It can also be observed from the experimental data of different processes that different process parameter combinations may lead to the same or similar performance indicators. For example, in turning, multiple combinations of process parameters may lead to the same or similar value of surface roughness. In EMM, multiple combinations of process parameters may lead to the same or similar value of taper and overcut. Therefore, there is a possibility to develop a method that can provide multiple optimal process parameter combinations for a single-objective optimization problem.
In this paper, the presented method is to obtain multiple optimal process parameter combinations for a single-objective optimization problem by splitting the original search space into smaller sub-search spaces and finding the optimal process parameter combinations in each sub-search space. Two different methods are used to split the original search space, and GA is utilized to optimize the process parameters in each sub-search space. The optimization results obtained after using the two search space splitting methods are compared to the optimization results obtained when the original search space was divided equally into smaller sub-search spaces; GA was used to optimize the process parameters in each sub-search space. EMM of SUS 304 is used as a case study because its experimental data shows that multiple process parameter combinations can lead to the same performance indicators. Due to the lack of physics-based models, a general regression neural network (GRNN) is used to create a forward prediction model between the input process parameters and the performance indicators for the process of EMM. The rest of the paper is organized as follows: section “Modeling” describes the modeling stage of the method. Section “Case study” presents and discusses the results obtained. Section “Conclusion” presents conclusions from the presented work and mentions future directions for the proposed approach.
Modeling
Split-optimization approach
Because the results obtained after using GA depend on the training accuracy of the GRNN, it is important to train the GRNN sufficiently so that it can predict the performance indicators with a high degree of accuracy. As there will always be some degree of error associated with the outputs of the GRNN, a possible method to cope with these errors is to take into consideration the significance level of the optimization problem. The significance level here is defined as a customized parameter that allows solutions with a fitness value better than or equal to it to be counted as final optimal solutions. The significance level by default is regarded as zero, which indicates that only solutions with the same minimum fitness value can be regarded as the final optimum solutions.
Splitting strategies
- (1)
Hill and valley splitting strategy
- a.
Identify two data points, A and B, from the experimental data set whose input values are furthest away from each other. Here, \(A = \left( {a_{1} ,a_{2} , \cdots ,a_{n} ,y_{a} } \right)\) and \(B = \left( {b_{1} ,b_{2} , \cdots ,b_{n} ,y_{b} } \right)\), indicating that all the data points have n inputs and 1 output.
- b.
Select a random data point C _{ 1 } from the remaining data points and determine whether it is a hill, valley, or neither compared to the initial points, i.e., A and B, based on the value of its output. For example, if \(y_{a}\) < \(y_{b}\) < \(y_{{c_{1} }}\), then C _{ 1 } is a hill; if \(y_{{c_{1} }}\) < \(y_{a}\) < \(y_{b}\), then C _{ 1 } is a valley; if \(y_{a}\) < \(y_{{c_{1} }}\) < \(y_{b}\), then C _{ 1 } is neither.
- c.
Select a random data point C _{ 2 } from the remaining data points; find a pair of previously selected data points whose input values encompass the input values of C _{ 2 }.
- d.
Compare the output value of C _{ 2 } with the data points selected in step c and determine whether it is a hill, valley, or neither.
- e.
Repeat step c and d until all the data points have been identified as a hill, valley, or neither.
- (2)
Cluster centers splitting strategy
GRNN-GA optimization
As mentioned earlier, a forward prediction model was created using GRNN (Specht 1991). The inputs of the GRNN were voltage, pulse on time, and feed rate; the outputs were D _{ in } and D _{ out }. During the training of the GRNN, the original data was split into training, validation, and testing data sets, and tenfold cross validation was used during the training phase of the GRNN to avoid overfitting and to find the optimal value of the spread parameter that would minimize the mean squared error (MSE). Once the GRNN was trained sufficiently, it was then utilized as the fitness function for GA during the optimization procedure.
Case study
Original range of the controllable process parameters
Process parameter | Voltage (V) | Pulse on time (µs) | Feed rate (µm/s) |
---|---|---|---|
Lower bound | 8 | 25 | 4 |
Upper bound | 20 | 70 | 12 |
Description of the case
Other basic information and settings are as follows: the electrolyte velocity was 10 m/s, electrolyte temperature was 27 °C, the initial gap between the tool and the workpiece was 100 µm, tool moving distance was 800 µm, the workpiece material was SUS 304, the electrolyte used was 10 %wt. NaNO_{3}, the nominal diameter of the hole was 900 µm, and the depth of the hole was 500 µm.
Voltage, pulse on time, and feed rate were used as the controllable process parameters, while the inner diameter of the micro-hole D _{ in } and the outer diameter D _{ out } were the measurable performances. The range of each process parameter is shown in Table 1. The range of the variables was fixed by taking into consideration two factors: 1. limitation of the devices used for EMM and 2. making sure that the experimental conditions would be stable within the chosen range. The resolution of the process parameters were was 0.1 V for the voltage, 0.1 µs for pulse on time, and 0.1 µm/s for the feed rate. This indicates that there are close to 3 million possible combinations of all the process parameters. Therefore, the proposed method was applied for this particular case study.
Levels of voltage, pulse on time, and feed rate values used for the three experimental sets
Experimental set # | Levels of voltage (V) | Levels pulse on time (µs) | Levels feed rate (µm/s) |
---|---|---|---|
1 | [16, 18, 20] | 25 | [4, 6, 8] |
2 | [4, 6, 8] | [50, 60, 70] | [8, 10, 12] |
3 | [4, 6, 8] | [50, 60, 70] | [8–10] |
The 63 groups of experimental data
No. | Voltage (V) | Pulse on time (µs) | Feed rate (µm/s) | D _{ in } (µm) | D _{ out } (µm) | Taper | Overcut (µm) |
---|---|---|---|---|---|---|---|
1 | 16 | 25 | 8 | 893 | 860 | 0.066 | 3.5 |
2 | 18 | 25 | 8 | 929 | 913 | 0.032 | 14.5 |
3 | 20 | 25 | 8 | 923 | 910 | 0.026 | 11.5 |
4 | 16 | 25 | 6 | 904 | 892 | 0.024 | 2 |
5 | 18 | 25 | 6 | 934 | 931 | 0.006 | 17 |
6 | 20 | 25 | 6 | 999 | 977 | 0.044 | 49.5 |
7 | 16 | 25 | 4 | 983 | 979 | 0.008 | 41.5 |
8 | 18 | 25 | 4 | 1050 | 1045 | 0.01 | 75 |
9 | 20 | 25 | 4 | 1125 | 1123 | 0.004 | 112.5 |
10 | 8 | 50 | 8 | 657.5 | 627.5 | 0.06 | 121.25 |
11 | 10 | 50 | 8 | 809.5 | 807.25 | 0.0045 | 45.25 |
12 | 12 | 50 | 8 | 866.25 | 858 | 0.0165 | 16.875 |
13 | 8 | 50 | 6 | 760 | 741 | 0.038 | 70 |
14 | 10 | 50 | 6 | 828.5 | 829.5 | 0.002 | 35.75 |
15 | 12 | 50 | 6 | 908.75 | 905.5 | 0.0065 | 4.375 |
16 | 8 | 50 | 4 | 781.75 | 780.25 | 0.003 | 59.125 |
17 | 10 | 50 | 4 | 887.25 | 881.75 | 0.011 | 6.375 |
18 | 12 | 50 | 4 | 957.75 | 970 | 0.0245 | 28.875 |
19 | 8 | 60 | 8 | 771.33 | 759.33 | 0.024 | 64.335 |
20 | 10 | 60 | 8 | 806.75 | 799.5 | 0.0145 | 46.625 |
21 | 12 | 60 | 8 | 862.75 | 847 | 0.0315 | 18.625 |
22 | 8 | 60 | 6 | 756.5 | 739.75 | 0.0335 | 71.75 |
23 | 10 | 60 | 6 | 776.75 | 777.5 | 0.0015 | 61.625 |
24 | 12 | 60 | 6 | 840.25 | 841.25 | 0.002 | 29.875 |
25 | 8 | 60 | 4 | 769 | 771.5 | 0.005 | 65.5 |
26 | 10 | 60 | 4 | 854.75 | 865.25 | 0.021 | 22.625 |
27 | 12 | 60 | 4 | 928.25 | 945.5 | 0.0345 | 14.125 |
28 | 8 | 70 | 8 | 718 | 721.5 | 0.007 | 91 |
29 | 10 | 70 | 8 | 779 | 796.75 | 0.0355 | 60.5 |
30 | 12 | 70 | 8 | 841.5 | 849.75 | 0.0165 | 29.25 |
31 | 8 | 70 | 6 | 736.5 | 744.5 | 0.016 | 81.75 |
32 | 10 | 70 | 6 | 802 | 829.75 | 0.0555 | 49 |
33 | 12 | 70 | 6 | 858.75 | 865 | 0.0125 | 20.625 |
34 | 8 | 70 | 4 | 783.25 | 783.25 | 0 | 58.375 |
35 | 10 | 70 | 4 | 878.75 | 872 | 0.0135 | 10.625 |
36 | 12 | 70 | 4 | 946.25 | 955.25 | 0.018 | 23.125 |
37 | 8 | 50 | 8 | 874 | 704 | 0.34 | 13 |
38 | 9 | 50 | 8 | 914 | 789 | 0.25 | 7 |
39 | 10 | 50 | 8 | 999 | 827 | 0.344 | 49.5 |
40 | 8 | 50 | 6 | 922 | 765 | 0.314 | 11 |
41 | 9 | 50 | 6 | 955 | 807 | 0.296 | 27.5 |
42 | 10 | 50 | 6 | 1039 | 837 | 0.404 | 69.5 |
43 | 8 | 50 | 4 | 932 | 797 | 0.27 | 16 |
44 | 9 | 50 | 4 | 1044 | 790 | 0.508 | 72 |
45 | 10 | 50 | 4 | 1130 | 858 | 0.544 | 115 |
46 | 8 | 60 | 8 | 903 | 708 | 0.39 | 1.5 |
47 | 9 | 60 | 8 | 967 | 766 | 0.402 | 33.5 |
48 | 10 | 60 | 8 | 1084 | 817 | 0.534 | 92 |
49 | 8 | 60 | 6 | 917 | 760 | 0.314 | 8.5 |
50 | 9 | 60 | 6 | 1043 | 856 | 0.374 | 71.5 |
51 | 10 | 60 | 6 | 1115 | 871 | 0.488 | 107.5 |
52 | 8 | 60 | 4 | 1071 | 754 | 0.634 | 85.5 |
53 | 9 | 60 | 4 | 1087 | 972 | 0.23 | 93.5 |
54 | 10 | 60 | 4 | 1263 | 1044 | 0.438 | 181.5 |
55 | 8 | 70 | 8 | 875 | 789 | 0.172 | 12.5 |
56 | 9 | 70 | 8 | 1071 | 842 | 0.458 | 85.5 |
57 | 10 | 70 | 8 | 1158 | 862 | 0.592 | 129 |
58 | 8 | 70 | 6 | 987 | 846 | 0.282 | 43.5 |
59 | 9 | 70 | 6 | 1212 | 886 | 0.652 | 156 |
60 | 10 | 70 | 6 | 1243 | 1056 | 0.374 | 171.5 |
61 | 8 | 70 | 4 | 1134 | 877 | 0.514 | 117 |
62 | 9 | 70 | 4 | 1260 | 935 | 0.65 | 180 |
63 | 10 | 70 | 4 | 1348 | 1016 | 0.664 | 224 |
Results
Parameter values used for GA
Number of generations | Population size | Crossover fraction | Mutation fraction | Elite count |
---|---|---|---|---|
100 | 50 | 0.85 | 0.15 | 3 |
- (1)
Hill and valley spitting strategy
Splitting result of the hill and valley splitting strategy
Number of hills | Number of valleys | Sub-range of voltage (V) | Sub-range of pulse on time (µs) | Sub-range of feed rate (µm/s) | Number of sub-search spaces |
---|---|---|---|---|---|
29 | 14 | [8, 9]; [9, 10]; [10, 18]; [18, 20] | [25,50]; [50,60]; [60,70] | [4, 6]; [6, 8] | 24 |
Optimization results obtained after using the hill and valley splitting strategy
No. | Voltage (V) | Pulse on Time (µs) | Feed rate (µm/s) | D _{ in } (µm) | D _{ out } (µm) | Taper | Overcut (µm) |
---|---|---|---|---|---|---|---|
1 | 18.0 | 60.0 | 6.0 | 901.05 | 892.68 | 0.02 | 0.52 |
2 | 19.4 | 60.0 | 5.9 | 900.00 | 895.82 | 0.01 | 0.00 |
3 | 18 | 61.6 | 6.0 | 901.59 | 894.02 | 0.02 | 0.80 |
4 | 19.9 | 67.1 | 5.9 | 900.00 | 898.96 | 0.00 | 0.00 |
5 | 18.0 | 60.0 | 6.2 | 900.43 | 891.84 | 0.02 | 0.21 |
6 | 19.6 | 59.6 | 6.1 | 900.00 | 895.94 | 0.01 | 0.00 |
7 | 18.0 | 62.6 | 6.3 | 901.38 | 893.94 | 0.01 | 0.69 |
8 | 20.0 | 68.6 | 6.1 | 900.00 | 899.24 | 0.00 | 0.00 |
- (2)
Cluster centers splitting strategy
Splitting result obtained using cluster centers strategy
Value of k | Cluster centers | Sub-range of voltage (V) | Sub-range of pulse on time (µs) | Sub-range of feed rate (µm/s) | Number of sub-spaces |
---|---|---|---|---|---|
2 | [10 V,60 µs,6 µm/s]; [18 V,25 µs,6 µm/s] | [8,10]; [10,18]; [18, 20] | [25, 60]; [60,70] | [4, 6]; [6, 8] | 12 |
3 | [10 V,55 µs,6 µm/s]; [10 V,70 µs,6 µm/s]; [18 V,25 µs,6 µm/s] | [8,10]; [10,18]; [18,20] | [25,55]; [55,70] | [4, 6]; [6, 8] | 12 |
4 | [10 V,60 µs,6 µm/s]; [10 V,70 µs,6 µm/s]; [18 V,25 µs,6 µm/s]; [10 V,50 µs,6 µm/s] | [8,10]; [10, 18]; [18,20] | [25,50]; [50,60]; [60,70] | [4,6]; [6,8] | 18 |
5 | [9 V,50 µs,5 µm/s]; [10 V,70 µs,6 µm/s]; [18 V,25 µs,6 µm/s]; [10 V,60 µs,6 µm/s]; [10 V,50 µs,8 µm/s] | [8, 9]; [9, 10]; [10, 18]; [18, 20] | [25,50]; [50,60]; [60,70] | [4,5]; [5,6]; [6, 8] | 36 |
6 | [9 V,50 µs,5 µm/s]; [10 V,50 µs,8 µm/s]; [10 V,70 µs,7 µm/s]; [9 V,70 µs,5 µm/s]; [10 V,60 µs,6 µm/s]; [18 V,25 µs,6 µm/s] | [8, 9]; [9, 10]; [10, 18]; [18, 20] | [25,50]; [50,60]; [60,70] | [4, 5]; [5, 6]; [6, 7]; [7, 8] | 48 |
The optimization results obtained using the cluster centers splitting strategy with k = 6
No. | Voltage (V) | Pulse on time (µs) | Feed rate (µm/s) | D _{ in } (µm) | D _{ out } (µm) | Taper | Overcut (µm) |
---|---|---|---|---|---|---|---|
1 | 18.0 | 59.7 | 6.0 | 901.01 | 892.49 | 0.02 | 0.50 |
2 | 19.2 | 59.1 | 5.9 | 900.00 | 894.85 | 0.01 | 0.00 |
3 | 18.0 | 61.6 | 6.0 | 901.58 | 893.98 | 0.02 | 0.79 |
4 | 19.9 | 67.6 | 6.0 | 900.00 | 899.03 | 0.00 | 0.00 |
5 | 18.0 | 60.0 | 6.2 | 900.44 | 891.85 | 0.02 | 0.22 |
6 | 19.8 | 60.0 | 6.2 | 900.00 | 895.87 | 0.01 | 0.00 |
7 | 18.0 | 61.7 | 6.2 | 901.00 | 893.27 | 0.02 | 0.50 |
8 | 20.0 | 68.1 | 6.0 | 900.00 | 899.19 | 0.00 | 0.00 |
9 | 18.0 | 70.0 | 7.5 | 900.00 | 890.99 | 0.02 | 0.00 |
10 | 18.9 | 70.0 | 7.0 | 899.97 | 895.46 | 0.01 | −0.01 |
- (3)
Equally splitting strategy
Splitting result of equally splitting strategy
Sub-range of voltage (V) | Sub-range of pulse on time (µs) | Sub-range of feed rate (µm/s) | Number of sub-spaces |
---|---|---|---|
[8, 11]; [11, 14]; [14, 17]; [17, 20] | [25,36]; [36,47]; [47,58]; [58,70] | [4, 5]; [5, 6]; [6, 7]; [7, 8] | 64 |
The optimization result obtained using the equally splitting strategy
No. | Voltage (V) | Pulse on time (µs) | Feed rate (µm/s) | D _{ in } (µm) | D _{ out } (µm) | Taper | Overcut (µm) |
---|---|---|---|---|---|---|---|
1 | 19.1 | 58.8 | 6.0 | 900.01 | 894.33 | 0.01 | 0.01 |
2 | 20.0 | 65.8 | 5.8 | 900.00 | 898.78 | 0.00 | 0.00 |
3 | 19.2 | 58.7 | 6.1 | 900.00 | 894.42 | 0.01 | 0.00 |
4 | 20.0 | 69.4 | 6.3 | 900.00 | 899.29 | 0.00 | 0.00 |
5 | 18.8 | 70.0 | 7.0 | 899.98 | 895.45 | 0.01 | −0.01 |
Comparison and analysis
Comparison of three splitting strategies
Name of splitting strategy | Hill and valley splitting | Cluster centers splitting | Equally splitting | ||||
---|---|---|---|---|---|---|---|
Value of parameter | – | k = 2 | k = 3 | k = 4 | k = 5 | k = 6 | 4 |
No. of sub-spaces | 24 | 12 | 12 | 18 | 36 | 48 | 64 |
No. of solutions | 8 | 8 | 4 | 8 | 8 | 10 | 5 |
Percentage of useful sub-spaces | 33.3 % | 66.7 % | 33.3 % | 44.4 % | 22.2 % | 20.8 % | 7.8 % |
It can be observed that the equally splitting strategy is the least efficient way because its percentage of useful sub-search spaces is the lowest (7.8 %). The efficiency of hill and valley splitting is fixed because it lacks any controllable parameters and because the sequence in which points are selected can affect their classification. It can be seen from Table 10 that there is a correlation between the efficiency of cluster centers splitting and the value of k. However, there is no clear understanding between the value of k and the efficiency of the method and there is also no guideline for selecting the optimal value of k.
Result of an additional validation experiment
Voltage (V) | Pulse on time (µs) | Feed rate (µm/s) | D _{ in } (µm) | D _{ out } (µm) | Taper | Overcut (µm) | |
---|---|---|---|---|---|---|---|
Predicted | 19.2 | 59.1 | 6 | 900.00 | 894.85 | 0.01 | 0.00 |
Experimental | 899 | 894 | 0.01 | 0.00 |
Based on the validation experimental result, it can be seen that the prediction error of the NN prediction model used in this case study is quite low and the results obtained using the proposed approach are better than the results shown in the initial experimental data. Noteworthy, the optimized input process parameter combination was not in the initial training dataset and the optimization algorithm was able to find a better-than-ever objective value. Therefore, the optimization result is verified.
Conclusion
In this paper, a split-optimization approach was proposed for obtaining multiple solutions for a single-objective process parameter optimization problem. The proposed approach consisted of two stages: splitting of the original search space into smaller sub-search spaces and optimization of process parameters in each of the smaller sub-search spaces. Two splitting strategies, i.e., hill and valley splitting strategy and cluster centers splitting strategy, were used to split the original search space into smaller sub-search spaces efficiently. Next, GA was used in each sub-search space to find multiple combinations of process parameters that minimized the single-objective value, one from each sub-search space. The efficiency of these two strategies was verified by comparing them with a method in which the original search space is divided into smaller and equal sub-search spaces. The comparison of the results from the different splitting methods showed that the hill and valley splitting strategy and cluster centers splitting strategy were more efficient than the equal splitting strategy. Among all the methods, the cluster centers splitting strategy, for a k value of 6, was able to achieve the most optimal solutions. The results obtained from the hill and valley splitting strategy showed that though it is an efficient method, its efficiency depends on the order in which the points are classified as a hill or valley.
Possible future work includes a study of the relationship between the efficiency of the cluster centers splitting strategy and the k value; a guideline should be to choose an optimal value of k. Future works also include experimentally validating the multiple solutions obtained using the proposed approach, applying the proposed approach to more case studies, and refining the proposed approach based on the results of the experimental validation and other case studies.
Declarations
Authors’ contributions
MR and ZP—analysis of data, development of the required code, and writing of the manuscript. YGY, ZWF, HYC, and WCW—study, collection and analysis of data. BL—comments for the paper. SYL—guideline for the proposed approach and comments for the paper. All authors read and approved the final manuscript.
Acknowledgements
We would like to acknowledge the Metal Industries Research & Development Center for collecting the data and providing us background knowledge regarding EMM.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
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