# Surface Performance Multiobjective Decision of a Cold Roll-Beating Spline with the Entropy Weight Ideal Point Method

Authors: F. K. Cui; F. Liu; Y. X. Su; X. L. Ruan; S. K. Xu; L. B. Liu
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Surface performance is an important indicator of the performance of cold roll-beating spline processing. To obtain the best cold roll spline surface performance (surface roughness, residual stress, and surface hardening degree), multiobjective optimal process parameters must be determined. To this end, this paper takes the cold roll-beating spline as the object of study and carries out a cold roll-beating spline surface performance test study. An ideal algorithm for entropy weight is constructed, and the multiobjective decision of the cold roll-beating spline surface performance is determined by using the entropy weight ideal point algorithm, providing a decision on the cold roll-beating spline processing parameters. The grey correlation algorithm is used for verification, and the results show that the multiobjective decision of the cold roll-beating spline surface performance is feasible by using the constructed entropy weight ideal point algorithm.


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## Body

## 1. Introduction

Cold roll-beating forming technology is an advanced plastic forming technology. Through high-speed rotation of the roller on the spline for continuous rolling and hitting, the metal material is forced to flow along the contour of the roller, gradually forming plastic; this approach is thus a green, nearly net forming, high-performance part processing method [1–3]. The technology has high production efficiency and low cost and requires no cutting or chip production. In the automotive industry, aerospace industry, major strategic equipment industry, and other fields that utilize a spline, gear and screw processing has been widely used [4, 5]. In the cold roll-beating forming process, the spline surface performance directly affects the use performance of the spline. Therefore, to obtain the best cold roll-beating spline surface performance, multiobjective decision research on the surface performance of a spline is performed to determine the performance of a cold roll-beating spline surface with improved processing indicator parameters, which has important theoretical significance and engineering application value for the processing of cold roll-beating splines.In terms of metal flow, Cui et al. [6] established the finite element model of spline in the process of cold roll forming according to the principle of cold roll forming and the law of minimum resistance and got the flow rule of each node in the process of cold roll forming. Cui et al. [7] established the contact model of rolling wheel and spline in the process of cold roll forming by the principle of spline cold rolling and obtained the flow path of nodes in the deformation area. In terms of work hardening, Cui et al. [8] explored the mechanism of work hardening on spline surface according to the principle of cold roll forming and made the cold rolling spline test to get the equivalent strain and temperature distribution rule during cold roll forming process. Cui et al. [9] based on the thermal effect generated by rotational speed of different roller wheels, a cold rolling finite element model was built. By analyzing the equivalent stress change caused by thermal effect, the influence rule of thermal effect on work hardening was obtained. In terms of design, Li et al. [10] improved the cold roll-beating spindle into a split-clamp structure to solve the problem of the frequent impact of spindle in forming process. Ma et al. [11] designed the improved structure of the hobbing wheel based on the interference of the hobbing wheel and the workpiece in the cold rolling forming process of continuous indexing. In terms of deformation force, Yang et al. [12] simulated the cold roll forming process with finite element software. By modifying the regression analysis formula, the deformation force of cold rolling was obtained under different technological parameters. Li et al. [13] established the finite element model of the cold rolling of the lead screw by analyzing the forming principle of the cold rolling beating of the screw and got the stress strain and the metal flow rule of the screw rolling cold rolling. In terms of microstructure evolution, Cui et al. [14] carried out the cold roll compression test according to the plastic deformation mechanism, established the relationship between macroscopic deformation and microscopic change, and got the parameters of 40 Cr. Cui et al. [15] obtained the Hopkinson compression test under different strain rates and obtained the dynamic response characteristics of the microstructure and the influence of microstructure change on the macroscopic properties of materials at different strain rates. In terms of residual stress, Ding et al. [16] took involute spline as the research object and used the contour method to measure the spatial distribution of residual stress and obtained the change rule between stress and depth.In summary, many scholars have performed a considerable amount of research on high-speed cold roll-beating experimentally and by studying metal flow, work hardening, design, deformation force, microstructure evolution, and residual stress. However, there are few reports on the decision of cold roll-beating spline processing parameters, and research on the multiobjective decision of cold roll-beating spline surface performance has not been sufficiently reported. Therefore, this paper proposes the use of the entropy weight ideal point method for the surface performance multiobjective decision of a cold roll-beating spline. Based on the results of a cold roll-beating spline test and using the normalized method, the weights of surface roughness, residual stress, and surface hardening are obtained by the entropy weight method. The ideal point method is then used to obtain the experimental order close to the ideal point of proximity and to determine the decision based on the cold roll-beating spline surface performance of the indicators using improved processing parameters, thereby enhancing the surface properties of the cold roll-beating spline.


## 2. Cold Roll-Beating Spline Test

### 2.1. Test Equipment

The Swiss company GROB’s ZRMe9 roll machine was utilized for the cold roll-beating spline test, and the surface roughness was measured using a Leica DCM3D white light copolymer interference microscope. The surface hardness was measured using a HVS-1000A microhardness tester, and the residual stress was measured by the contour method.


### 2.2. Experiment Material

Spline rough material was used for grade 20 steel, with a normalizing treatment at 910°C. The main chemical composition of grade 20 steel is shown in Table1.Table 1
Major chemical composition of grade 20 steel (mass fraction, %).
C Si Mn P S Ni Cr  0.20 0.17~0.37 0.35~0.65 ≤0.035 ≤0.035 ≤0.30 ≤0.25


### 2.3. Test Parameters

The cold roll-beating spline has a spool diameter of 35.15 mm, a modulus of 2.5, 14 teeth, a pressure angle of 30°, a tooth high coefficient of 0.5, and a tooth root high coefficient of 0.7. The parameters of the cold roll-beating forming process are as follows: the cold roll-beating speedn is 1428, 1581, 1806, 2032, and 2258 r/min, while the feed rate f1 is 21, 28, 35, and 42 mm/min.


### 2.4. Experimental Procedure

#### 2.4.1. Spline Cold Roll-Beating Forming

The ZRMe9 roll machine uses the pull-out method to start the cold roll-beating involute spline and roll-beating wheel symmetrical installation. According to the principle of the cold roll-beating spline [17], when the wheel rotates 14 turns, the workpiece rotates one turn; thus, in order for the roll of the workpiece to hit the same spline slot, the workpiece feed f should be 7f1/n.


#### 2.4.2. Surface Roughness Test Process

Using a wire cutting machine, the splines selected as samples were taken,a=10 mm, b=2 mm, and c=6.79 mm; one sample is shown in Figure 1. The samples were placed under a Leica DCM3D white light copolymer interference microscope, and the measurement area on the sample was selected to be 1.27 mm × 0.42 mm, with the magnification of the microscope set at 250x. To ensure the accuracy of the measurement results, three different parts at each round of the sample were measured separately, and the measured surface profile data were Gaussian-filtered and then separated from the surface roughness data and waviness data. Finally, the average value was obtained and used as the surface roughness value of the sample.Figure 1
Spline sample.


#### 2.4.3. Residual Stress Measurement Process [16, 18–20]

A walking wire was used to cut the machine, cutting a tooth from the cold roll-beating spline (shown in Figure2) using bronze wire with a diameter of 0.1 mm at a feed rate of 2 mm/min with a feed rate of 0.5 mm/min along the shadow part of the spline, as shown in Figure 2. A sample was symmetrically cut, with l=10 mm, w=4.35 mm, and h0=2.68 mm. The contact points on the two sections of the sample shown in Figure 1 were measured using a contact coordinate measuring instrument with a precision of 0.1 μm and measurement point spacing of 0.01 mm × 0.01 mm. The measured contour data of the two sections were averaged, and the data are fitted with a smooth response surface. The fitted contour surface was applied as a boundary condition to the finite element model established by the spline tooth size obtained by the annealing process. For the finite element calculation, the stress on the cut surface of the obtained annealed sample was equivalent to the residual stress at the same position before the specimen was cut. The use of contour method to measure residual stress steps is very complicated. Even if each step is carried out in strict accordance with operational requirements, there are still some unavoidable factors that interfere with the experimental results. The main error of the contour method in measuring the residual stress is the value of the measured residual stress; the values will generally be slightly smaller during pretreatment release and may be negligible. When cutting the sample, there will be some changes in the stress distribution in the cutting area and we cut cutting speed and discharge current while cutting to avoid the cutting process to maximize the impact of cutting temperature area. In addition, our research group used the crack compliance method and the XRD residual stress measurement method to carry out the related experiments; the results show that the use of contouring method to measure residual stress is found to be more stable. In fact, in order to ensure that the measurement results are more reasonable and rigorous, we use a simulation model based on experimental data for a large number of comparative analyses of this issue. There is a certain degree of deviation between experimental results and simulation results, but the deviation is very small.Figure 2
The shape of the specimen and the position of the cross section to be measured.


#### 2.4.4. Hardening Degree Test Process

The cold roll-beating splines were pulled on the roll machine. The wire cutting machine was used along the direction of the spline to cut into a section of the tooth (as shown in Figure1). The cut samples were then ground and polished. The HVS-1000 N microhardness tester was used to measure the Vickers hardness at the index circle along the normal surface, and three points at different positions at the same level are measured on this part, with a distance of 0.1 mm between each measuring point. The load is 1 N for each measurement and the loading time is 10 s, taking the average of three measurements as the microhardness at this level. Because the degree of hardening is an important indicator when evaluating work hardening, the test results for the surface layer are represented after processing with the microhardness of the surface layer before the microhardness ratio.


### 2.5. Test Results

The cold roll-beating spline test machining parameters are the roller speed and feed rate. The test results for the tooth surface roughness, the residual stress on the surface of the tooth surface, and the degree of hardening are shown in Table2.Table 2
Cold roll-beating processing test results.
Test order Rolling wheel speed (r/min) Feed ratef1 (mm/min) Feed ratef (mm/r) Surface roughness (μm) Residual stress (MPa) Degree of hardening (%) 1 1428 21 0.103 0.469 ± 0.0032 - 67.42 ± 6.35 140.96 ± 2.26 2 1428 28 0.137 0.479 ± 0.0028 - 72.70 ± 4.25 143.35 ± 2.14 3 1428 35 0.172 0.530 ± 0.0017 - 79.38 ± 3.36 145.83 ± 1.97 4 1428 42 0.206 0.625 ± 0.0006 - 83.01 ± 2.58 148.71 ± 1.64 5 1581 21 0.093 0.440 ± 0.0034 - 68.76 ± 5.47 140.25 ± 2.31 6 1581 28 0.124 0.449 ± 0.0027 - 75.16 ± 4.42 143.06 ± 2.05 7 1581 35 0.155 0.467 ± 0.0018 - 78.79 ± 3.86 144.91 ± 1.93 8 1581 42 0.186 0.508 ± 0.0012 - 83.11 ± 2.71 146.79 ± 1.76 9 1806 21 0.081 0.383 ± 0.0037 - 72.15 ± 5.13 138.78 ± 2.35 10 1806 28 0.109 0.397 ± 0.0029 - 75.65 ± 4.65 142.26 ± 2.01 11 1806 35 0.136 0.442 ± 0.0016 - 79.49 ± 3.03 143.48 ± 1.84 12 1806 42 0.163 0.495 ± 0.0014 - 84.83 ± 1.75 145.95 ± 1.72 13 2032 21 0.072 0.345 ± 0.0033 - 73.04 ± 5.56 138.41 ± 2.39 14 2032 28 0.096 0.361 ± 0.0024 - 79.04 ± 4.16 141.43 ± 2.22 15 2032 35 0.121 0.433 ± 0.0016 - 82.77 ± 3.23 142.64 ± 1.89 16 2032 42 0.145 0.500 ± 0.0007 - 84.76 ± 2.53 145.03 ± 1.59 17 2258 21 0.065 0.357 ± 0.0033 - 74.75 ± 5.96 138.11 ± 2.43 18 2258 28 0.087 0.368 ± 0.0023 - 78.58 ± 4.64 139.37 ± 2.18 19 2258 35 0.109 0.411 ± 0.0017 - 83.25 ± 2.28 141.93 ± 1.97 20 2258 42 0.130 0.472 ± 0.0009 - 84.87 ± 1.82 144.11 ± 1.81


## 2.1. Test Equipment

The Swiss company GROB’s ZRMe9 roll machine was utilized for the cold roll-beating spline test, and the surface roughness was measured using a Leica DCM3D white light copolymer interference microscope. The surface hardness was measured using a HVS-1000A microhardness tester, and the residual stress was measured by the contour method.


## 2.2. Experiment Material

Spline rough material was used for grade 20 steel, with a normalizing treatment at 910°C. The main chemical composition of grade 20 steel is shown in Table1.Table 1
Major chemical composition of grade 20 steel (mass fraction, %).
C Si Mn P S Ni Cr  0.20 0.17~0.37 0.35~0.65 ≤0.035 ≤0.035 ≤0.30 ≤0.25


## 2.3. Test Parameters

The cold roll-beating spline has a spool diameter of 35.15 mm, a modulus of 2.5, 14 teeth, a pressure angle of 30°, a tooth high coefficient of 0.5, and a tooth root high coefficient of 0.7. The parameters of the cold roll-beating forming process are as follows: the cold roll-beating speedn is 1428, 1581, 1806, 2032, and 2258 r/min, while the feed rate f1 is 21, 28, 35, and 42 mm/min.


## 2.4. Experimental Procedure

### 2.4.1. Spline Cold Roll-Beating Forming

The ZRMe9 roll machine uses the pull-out method to start the cold roll-beating involute spline and roll-beating wheel symmetrical installation. According to the principle of the cold roll-beating spline [17], when the wheel rotates 14 turns, the workpiece rotates one turn; thus, in order for the roll of the workpiece to hit the same spline slot, the workpiece feed f should be 7f1/n.


### 2.4.2. Surface Roughness Test Process

Using a wire cutting machine, the splines selected as samples were taken,a=10 mm, b=2 mm, and c=6.79 mm; one sample is shown in Figure 1. The samples were placed under a Leica DCM3D white light copolymer interference microscope, and the measurement area on the sample was selected to be 1.27 mm × 0.42 mm, with the magnification of the microscope set at 250x. To ensure the accuracy of the measurement results, three different parts at each round of the sample were measured separately, and the measured surface profile data were Gaussian-filtered and then separated from the surface roughness data and waviness data. Finally, the average value was obtained and used as the surface roughness value of the sample.Figure 1
Spline sample.


### 2.4.3. Residual Stress Measurement Process [16, 18–20]

A walking wire was used to cut the machine, cutting a tooth from the cold roll-beating spline (shown in Figure2) using bronze wire with a diameter of 0.1 mm at a feed rate of 2 mm/min with a feed rate of 0.5 mm/min along the shadow part of the spline, as shown in Figure 2. A sample was symmetrically cut, with l=10 mm, w=4.35 mm, and h0=2.68 mm. The contact points on the two sections of the sample shown in Figure 1 were measured using a contact coordinate measuring instrument with a precision of 0.1 μm and measurement point spacing of 0.01 mm × 0.01 mm. The measured contour data of the two sections were averaged, and the data are fitted with a smooth response surface. The fitted contour surface was applied as a boundary condition to the finite element model established by the spline tooth size obtained by the annealing process. For the finite element calculation, the stress on the cut surface of the obtained annealed sample was equivalent to the residual stress at the same position before the specimen was cut. The use of contour method to measure residual stress steps is very complicated. Even if each step is carried out in strict accordance with operational requirements, there are still some unavoidable factors that interfere with the experimental results. The main error of the contour method in measuring the residual stress is the value of the measured residual stress; the values will generally be slightly smaller during pretreatment release and may be negligible. When cutting the sample, there will be some changes in the stress distribution in the cutting area and we cut cutting speed and discharge current while cutting to avoid the cutting process to maximize the impact of cutting temperature area. In addition, our research group used the crack compliance method and the XRD residual stress measurement method to carry out the related experiments; the results show that the use of contouring method to measure residual stress is found to be more stable. In fact, in order to ensure that the measurement results are more reasonable and rigorous, we use a simulation model based on experimental data for a large number of comparative analyses of this issue. There is a certain degree of deviation between experimental results and simulation results, but the deviation is very small.Figure 2
The shape of the specimen and the position of the cross section to be measured.


### 2.4.4. Hardening Degree Test Process

The cold roll-beating splines were pulled on the roll machine. The wire cutting machine was used along the direction of the spline to cut into a section of the tooth (as shown in Figure1). The cut samples were then ground and polished. The HVS-1000 N microhardness tester was used to measure the Vickers hardness at the index circle along the normal surface, and three points at different positions at the same level are measured on this part, with a distance of 0.1 mm between each measuring point. The load is 1 N for each measurement and the loading time is 10 s, taking the average of three measurements as the microhardness at this level. Because the degree of hardening is an important indicator when evaluating work hardening, the test results for the surface layer are represented after processing with the microhardness of the surface layer before the microhardness ratio.


## 2.4.1. Spline Cold Roll-Beating Forming

The ZRMe9 roll machine uses the pull-out method to start the cold roll-beating involute spline and roll-beating wheel symmetrical installation. According to the principle of the cold roll-beating spline [17], when the wheel rotates 14 turns, the workpiece rotates one turn; thus, in order for the roll of the workpiece to hit the same spline slot, the workpiece feed f should be 7f1/n.


## 2.4.2. Surface Roughness Test Process

Using a wire cutting machine, the splines selected as samples were taken,a=10 mm, b=2 mm, and c=6.79 mm; one sample is shown in Figure 1. The samples were placed under a Leica DCM3D white light copolymer interference microscope, and the measurement area on the sample was selected to be 1.27 mm × 0.42 mm, with the magnification of the microscope set at 250x. To ensure the accuracy of the measurement results, three different parts at each round of the sample were measured separately, and the measured surface profile data were Gaussian-filtered and then separated from the surface roughness data and waviness data. Finally, the average value was obtained and used as the surface roughness value of the sample.Figure 1
Spline sample.


## 2.4.3. Residual Stress Measurement Process [16, 18–20]

A walking wire was used to cut the machine, cutting a tooth from the cold roll-beating spline (shown in Figure2) using bronze wire with a diameter of 0.1 mm at a feed rate of 2 mm/min with a feed rate of 0.5 mm/min along the shadow part of the spline, as shown in Figure 2. A sample was symmetrically cut, with l=10 mm, w=4.35 mm, and h0=2.68 mm. The contact points on the two sections of the sample shown in Figure 1 were measured using a contact coordinate measuring instrument with a precision of 0.1 μm and measurement point spacing of 0.01 mm × 0.01 mm. The measured contour data of the two sections were averaged, and the data are fitted with a smooth response surface. The fitted contour surface was applied as a boundary condition to the finite element model established by the spline tooth size obtained by the annealing process. For the finite element calculation, the stress on the cut surface of the obtained annealed sample was equivalent to the residual stress at the same position before the specimen was cut. The use of contour method to measure residual stress steps is very complicated. Even if each step is carried out in strict accordance with operational requirements, there are still some unavoidable factors that interfere with the experimental results. The main error of the contour method in measuring the residual stress is the value of the measured residual stress; the values will generally be slightly smaller during pretreatment release and may be negligible. When cutting the sample, there will be some changes in the stress distribution in the cutting area and we cut cutting speed and discharge current while cutting to avoid the cutting process to maximize the impact of cutting temperature area. In addition, our research group used the crack compliance method and the XRD residual stress measurement method to carry out the related experiments; the results show that the use of contouring method to measure residual stress is found to be more stable. In fact, in order to ensure that the measurement results are more reasonable and rigorous, we use a simulation model based on experimental data for a large number of comparative analyses of this issue. There is a certain degree of deviation between experimental results and simulation results, but the deviation is very small.Figure 2
The shape of the specimen and the position of the cross section to be measured.


## 2.4.4. Hardening Degree Test Process

The cold roll-beating splines were pulled on the roll machine. The wire cutting machine was used along the direction of the spline to cut into a section of the tooth (as shown in Figure1). The cut samples were then ground and polished. The HVS-1000 N microhardness tester was used to measure the Vickers hardness at the index circle along the normal surface, and three points at different positions at the same level are measured on this part, with a distance of 0.1 mm between each measuring point. The load is 1 N for each measurement and the loading time is 10 s, taking the average of three measurements as the microhardness at this level. Because the degree of hardening is an important indicator when evaluating work hardening, the test results for the surface layer are represented after processing with the microhardness of the surface layer before the microhardness ratio.


## 2.5. Test Results

The cold roll-beating spline test machining parameters are the roller speed and feed rate. The test results for the tooth surface roughness, the residual stress on the surface of the tooth surface, and the degree of hardening are shown in Table2.Table 2
Cold roll-beating processing test results.
Test order Rolling wheel speed (r/min) Feed ratef1 (mm/min) Feed ratef (mm/r) Surface roughness (μm) Residual stress (MPa) Degree of hardening (%) 1 1428 21 0.103 0.469 ± 0.0032 - 67.42 ± 6.35 140.96 ± 2.26 2 1428 28 0.137 0.479 ± 0.0028 - 72.70 ± 4.25 143.35 ± 2.14 3 1428 35 0.172 0.530 ± 0.0017 - 79.38 ± 3.36 145.83 ± 1.97 4 1428 42 0.206 0.625 ± 0.0006 - 83.01 ± 2.58 148.71 ± 1.64 5 1581 21 0.093 0.440 ± 0.0034 - 68.76 ± 5.47 140.25 ± 2.31 6 1581 28 0.124 0.449 ± 0.0027 - 75.16 ± 4.42 143.06 ± 2.05 7 1581 35 0.155 0.467 ± 0.0018 - 78.79 ± 3.86 144.91 ± 1.93 8 1581 42 0.186 0.508 ± 0.0012 - 83.11 ± 2.71 146.79 ± 1.76 9 1806 21 0.081 0.383 ± 0.0037 - 72.15 ± 5.13 138.78 ± 2.35 10 1806 28 0.109 0.397 ± 0.0029 - 75.65 ± 4.65 142.26 ± 2.01 11 1806 35 0.136 0.442 ± 0.0016 - 79.49 ± 3.03 143.48 ± 1.84 12 1806 42 0.163 0.495 ± 0.0014 - 84.83 ± 1.75 145.95 ± 1.72 13 2032 21 0.072 0.345 ± 0.0033 - 73.04 ± 5.56 138.41 ± 2.39 14 2032 28 0.096 0.361 ± 0.0024 - 79.04 ± 4.16 141.43 ± 2.22 15 2032 35 0.121 0.433 ± 0.0016 - 82.77 ± 3.23 142.64 ± 1.89 16 2032 42 0.145 0.500 ± 0.0007 - 84.76 ± 2.53 145.03 ± 1.59 17 2258 21 0.065 0.357 ± 0.0033 - 74.75 ± 5.96 138.11 ± 2.43 18 2258 28 0.087 0.368 ± 0.0023 - 78.58 ± 4.64 139.37 ± 2.18 19 2258 35 0.109 0.411 ± 0.0017 - 83.25 ± 2.28 141.93 ± 1.97 20 2258 42 0.130 0.472 ± 0.0009 - 84.87 ± 1.82 144.11 ± 1.81


## 3. Multiobjective Decision Algorithm for Entropy Weighted Ideal Point

From the cold roll-beating spline test results, we can see that the surface roughness, residual stress, and surface hardening of the cold roll-beating spline surface performance for each index differ in each dimension, that multiple indicators affect the surface performance of the cold roll-beating spline, and that each indicator has a different degree of impact on the surface properties of the cold roll-beating spline. The cold roll-beating spline test results are impacted by so many features that it is difficult to find an improved decision for the cold roll-beating spline surface performance indicators of the processing parameters based solely on the test results of the cold roll-beating spline. Therefore, combining a limited number of evaluation objects and an idealized goal of the degree of proximity to evaluate the advantages and disadvantages of each object, a multiobjective decision algorithm for entropy weight ideal points is proposed. The multiobjective decision algorithm for the entropy weight ideal points is shown in Figure3.Figure 3
Flowchart of the multiobjective decision algorithm for entropy weight ideal points.


## 4. Research on the Surface Performance Multiobjective Decision of a Cold Roll-Beating Spline with the Entropy Weight Ideal Point Method

### 4.1. Normalized Method

In Table2, the dimensions of roughness, residual stress, and surface hardening are different in the three columns of data. To normalize the three datasets,(1)Pij=rij∑i=1mriji=1,2,…,20;j=5,6,7,where Pij is the result of normalization for the ith row and jth column and rij presents test result data for the ith row and jth column.Applying formula (1) to the results in Table 2 for the surface roughness, residual stress, and surface hardening, we can normalize the results obtained by the cold roll-beating spline surface performance indicators of the data normalization matrix V:(2)V=0.052510.043180.049370.053630.046560.050210.059340.050840.051080.069980.053160.052090.049270.044030.049130.050270.048130.050110.052290.050460.050760.056880.053220.051420.042880.046210.048470.044450.048450.049830.049490.050910.050260.055430.054330.051120.038630.046780.048480.040420.050620.049540.048480.053010.049960.055980.054280.050810.039970.047870.048380.041210.050320.048820.046020.053310.049710.052850.054350.05048.


### 4.2. Entropy Weight Method

Entropy weighting is an objective weighting method based on the entropy weight method, where the entropy weightwj is obtained indirectly from the entropy ej. Formula (3) is used to find the entropy ej, while formula (4) is used to find the entropy weight wj:(3)ej=-k∑i=1mvij∗lnvijk=1lnm,m=20,(4)wj=1-ej∑j=1n1-ej,where ej is the entropy of each index, k is the entropy coefficient, vij is the normalized data point in the ith row andjth column of V, and wj is the entropy weight of each index.Substituting the data inV in formula (3), the entropy value ej of the surface roughness, residual stress, and surface hardening degree is obtained. Substituting the entropy value ej into formula (4), the entropy weight wj of the surface roughness, residual stress, and surface hardening degree is obtained. The entropy value and entropy weights of the surface properties are shown in Table 3.Table 3
Surface performance of the indicators of entropy value and entropy weight.
Index Surface roughness Residual stress Degree of hardening Entropy value 0.9963 0.9992 0.9999 Entropy weight 0.8043 0.1739 0.0217


### 4.3. Ideal Point Method

#### 4.3.1. Weighted Normalization Matrix

The entropy weights in Table3 are multiplied by the respective indices corresponding to the data normalization matrix V, and the weighted normalization matrix V∗ of the surface performance indicators is obtained:(5)V∗=0.042240.007510.001070.043140.008100.001090.047730.008840.001110.056290.009240.001130.039630.007660.001070.040440.008370.001090.042060.008780.001100.045750.009260.001120.034490.008040.001050.035750.008420.001080.039810.008850.001090.044580.009450.001110.031070.008130.001050.032510.008800.001070.038990.009220.001080.045030.009440.001100.032150.008320.001050.033140.008750.001060.037010.009270.001080.042510.009450.00120.By weighting the normalization matrixV∗, one can evaluate the surface performance of the indicators of the maximum and minimum values. The ideal solution Z+ for the surface performance of the indicators corresponds to the largest maximum, while the negative ideal solution Z- for the surface performance of the indicators corresponds to the smallest minimum. To this end, we obtain the positive and negative ideal solutions of the surface performance indices:(6)Z+=0.05629,0.00945,0.00113,Z-=0.03107,0.00751,0.00105.


#### 4.3.2. Distance between the Surface Performance Indicators and the Positive and Negative Ideal Solutions

Using the weighted normalization matrixV∗ and the weighted normalization matrix V∗ corresponding to the positive ideal solution Z+ and the negative ideal solution Z-, formula (7) gives the distance Di+ between the data in the weighted normalization matrix V∗ and the positive ideal solution Z+. Using formula (8), the distance Di- between the data in the weighted normalization matrix V∗ and the negative ideal solution Z- is obtained:(7)Di+=∑j=1mVij∗-Zj+2i=1,2,…,n;j=1,2,3,(8)Di-=∑j=1mVij∗-Zj-2i=1,2,…,n;j=1,2,3,where Vij∗ is the element of the ith row and jth column, Zj+ is the ideal solution of the jth column, and Zj- is the negative ideal solution for column j.


### 4.4. Proximity

From formulas (7) and (8), the distance between the data and the ideal solution in the weighted normalized matrix V∗ is obtained, as is the distance to the negative ideal solution. Using formula (9), the proximity of the test order to the ideal point is obtained:(9)Ci=Di-Di++Di-i=1,2,…,n,where Ci is the ith proximity.


### 4.5. Multiobjective Decision for the Cold Roll-Beating Spline Surface Performance

From formulas (7) and (8), one can obtain the surface performance of the indicators and the distance between the positive and negative ideal solutions. The experimental order obtained by formula (9) is close to the proximity of the ideal point. The results are shown in Table 4.Table 4
Surface performance of the indicators, the distance between the positive and negative ideal solutions, and the test order close to the ideal point of proximity.
Test order D + D - C i 1 0.01419 0.01117 0.44 2 0.01322 0.01208 0.48 3 0.00858 0.01671 0.66 4 0.00022 0.02528 0.99 5 0.01676 0.00856 0.34 6 0.01589 0.009405 0.37 7 0.01425 0.01106 0.44 8 0.01054 0.01478 0.58 9 0.02184 0.00346 0.14 10 0.02056 0.00477 0.19 11 0.01650 0.00884 0.35 12 0.01171 0.01365 0.54 13 0.02526 0.00062 0.02 14 0.02379 0.00194 0.08 15 0.01730 0.00081 0.32 16 0.01126 0.01409 0.56 17 0.02417 0.00135 0.05 18 0.02316 0.00241 0.09 19 0.01928 0.00620 0.24 20 0.01378 0.01160 0.46The relationship between the test order shown in Figure4 and the corresponding proximity is obtained from Table 4.Figure 4
Cold roll-beating spline test order and the corresponding proximity.As shown in Figure4, the corresponding proximity of test order 4 is the best of all the proximities. As shown in Table 2, for test order 4, the corresponding roller speed is 1428 r/min and the feed rate is 0.206 mm/r (42 mm/min).


### 4.6. Verification of the Multiobjective Decision for the Cold Roll-Beating Spline Surface Performance

The grey correlation algorithm was used to calculate the grey correlative degree of the surface roughness, residual stress, and surface hardening degree of the surface performance, based on the order of the test and the grey correlation degree of the size, as shown in Figure5.Figure 5
Cold roll-beating spline test order and the corresponding grey correlation degree.Figure5 shows that test order 4 has a large grey correlation degree, as indicated in Table 2. For test order 4, the corresponding roller speed is 1428 r/min, and the feed rate is 0.206 mm/r (42 mm/min).Using the entropy weight ideal point method, an improved proximity is obtained for test order 4, and the grey correlation algorithm verifies that test order 4 has a large grey correlation degree, allowing better determination of the cold roll-beating spline processing parameters.


## 4.1. Normalized Method

In Table2, the dimensions of roughness, residual stress, and surface hardening are different in the three columns of data. To normalize the three datasets,(1)Pij=rij∑i=1mriji=1,2,…,20;j=5,6,7,where Pij is the result of normalization for the ith row and jth column and rij presents test result data for the ith row and jth column.Applying formula (1) to the results in Table 2 for the surface roughness, residual stress, and surface hardening, we can normalize the results obtained by the cold roll-beating spline surface performance indicators of the data normalization matrix V:(2)V=0.052510.043180.049370.053630.046560.050210.059340.050840.051080.069980.053160.052090.049270.044030.049130.050270.048130.050110.052290.050460.050760.056880.053220.051420.042880.046210.048470.044450.048450.049830.049490.050910.050260.055430.054330.051120.038630.046780.048480.040420.050620.049540.048480.053010.049960.055980.054280.050810.039970.047870.048380.041210.050320.048820.046020.053310.049710.052850.054350.05048.


## 4.2. Entropy Weight Method

Entropy weighting is an objective weighting method based on the entropy weight method, where the entropy weightwj is obtained indirectly from the entropy ej. Formula (3) is used to find the entropy ej, while formula (4) is used to find the entropy weight wj:(3)ej=-k∑i=1mvij∗lnvijk=1lnm,m=20,(4)wj=1-ej∑j=1n1-ej,where ej is the entropy of each index, k is the entropy coefficient, vij is the normalized data point in the ith row andjth column of V, and wj is the entropy weight of each index.Substituting the data inV in formula (3), the entropy value ej of the surface roughness, residual stress, and surface hardening degree is obtained. Substituting the entropy value ej into formula (4), the entropy weight wj of the surface roughness, residual stress, and surface hardening degree is obtained. The entropy value and entropy weights of the surface properties are shown in Table 3.Table 3
Surface performance of the indicators of entropy value and entropy weight.
Index Surface roughness Residual stress Degree of hardening Entropy value 0.9963 0.9992 0.9999 Entropy weight 0.8043 0.1739 0.0217


## 4.3. Ideal Point Method

### 4.3.1. Weighted Normalization Matrix

The entropy weights in Table3 are multiplied by the respective indices corresponding to the data normalization matrix V, and the weighted normalization matrix V∗ of the surface performance indicators is obtained:(5)V∗=0.042240.007510.001070.043140.008100.001090.047730.008840.001110.056290.009240.001130.039630.007660.001070.040440.008370.001090.042060.008780.001100.045750.009260.001120.034490.008040.001050.035750.008420.001080.039810.008850.001090.044580.009450.001110.031070.008130.001050.032510.008800.001070.038990.009220.001080.045030.009440.001100.032150.008320.001050.033140.008750.001060.037010.009270.001080.042510.009450.00120.By weighting the normalization matrixV∗, one can evaluate the surface performance of the indicators of the maximum and minimum values. The ideal solution Z+ for the surface performance of the indicators corresponds to the largest maximum, while the negative ideal solution Z- for the surface performance of the indicators corresponds to the smallest minimum. To this end, we obtain the positive and negative ideal solutions of the surface performance indices:(6)Z+=0.05629,0.00945,0.00113,Z-=0.03107,0.00751,0.00105.


### 4.3.2. Distance between the Surface Performance Indicators and the Positive and Negative Ideal Solutions

Using the weighted normalization matrixV∗ and the weighted normalization matrix V∗ corresponding to the positive ideal solution Z+ and the negative ideal solution Z-, formula (7) gives the distance Di+ between the data in the weighted normalization matrix V∗ and the positive ideal solution Z+. Using formula (8), the distance Di- between the data in the weighted normalization matrix V∗ and the negative ideal solution Z- is obtained:(7)Di+=∑j=1mVij∗-Zj+2i=1,2,…,n;j=1,2,3,(8)Di-=∑j=1mVij∗-Zj-2i=1,2,…,n;j=1,2,3,where Vij∗ is the element of the ith row and jth column, Zj+ is the ideal solution of the jth column, and Zj- is the negative ideal solution for column j.


## 4.3.1. Weighted Normalization Matrix

The entropy weights in Table3 are multiplied by the respective indices corresponding to the data normalization matrix V, and the weighted normalization matrix V∗ of the surface performance indicators is obtained:(5)V∗=0.042240.007510.001070.043140.008100.001090.047730.008840.001110.056290.009240.001130.039630.007660.001070.040440.008370.001090.042060.008780.001100.045750.009260.001120.034490.008040.001050.035750.008420.001080.039810.008850.001090.044580.009450.001110.031070.008130.001050.032510.008800.001070.038990.009220.001080.045030.009440.001100.032150.008320.001050.033140.008750.001060.037010.009270.001080.042510.009450.00120.By weighting the normalization matrixV∗, one can evaluate the surface performance of the indicators of the maximum and minimum values. The ideal solution Z+ for the surface performance of the indicators corresponds to the largest maximum, while the negative ideal solution Z- for the surface performance of the indicators corresponds to the smallest minimum. To this end, we obtain the positive and negative ideal solutions of the surface performance indices:(6)Z+=0.05629,0.00945,0.00113,Z-=0.03107,0.00751,0.00105.


## 4.3.2. Distance between the Surface Performance Indicators and the Positive and Negative Ideal Solutions

Using the weighted normalization matrixV∗ and the weighted normalization matrix V∗ corresponding to the positive ideal solution Z+ and the negative ideal solution Z-, formula (7) gives the distance Di+ between the data in the weighted normalization matrix V∗ and the positive ideal solution Z+. Using formula (8), the distance Di- between the data in the weighted normalization matrix V∗ and the negative ideal solution Z- is obtained:(7)Di+=∑j=1mVij∗-Zj+2i=1,2,…,n;j=1,2,3,(8)Di-=∑j=1mVij∗-Zj-2i=1,2,…,n;j=1,2,3,where Vij∗ is the element of the ith row and jth column, Zj+ is the ideal solution of the jth column, and Zj- is the negative ideal solution for column j.


## 4.4. Proximity

From formulas (7) and (8), the distance between the data and the ideal solution in the weighted normalized matrix V∗ is obtained, as is the distance to the negative ideal solution. Using formula (9), the proximity of the test order to the ideal point is obtained:(9)Ci=Di-Di++Di-i=1,2,…,n,where Ci is the ith proximity.


## 4.5. Multiobjective Decision for the Cold Roll-Beating Spline Surface Performance

From formulas (7) and (8), one can obtain the surface performance of the indicators and the distance between the positive and negative ideal solutions. The experimental order obtained by formula (9) is close to the proximity of the ideal point. The results are shown in Table 4.Table 4
Surface performance of the indicators, the distance between the positive and negative ideal solutions, and the test order close to the ideal point of proximity.
Test order D + D - C i 1 0.01419 0.01117 0.44 2 0.01322 0.01208 0.48 3 0.00858 0.01671 0.66 4 0.00022 0.02528 0.99 5 0.01676 0.00856 0.34 6 0.01589 0.009405 0.37 7 0.01425 0.01106 0.44 8 0.01054 0.01478 0.58 9 0.02184 0.00346 0.14 10 0.02056 0.00477 0.19 11 0.01650 0.00884 0.35 12 0.01171 0.01365 0.54 13 0.02526 0.00062 0.02 14 0.02379 0.00194 0.08 15 0.01730 0.00081 0.32 16 0.01126 0.01409 0.56 17 0.02417 0.00135 0.05 18 0.02316 0.00241 0.09 19 0.01928 0.00620 0.24 20 0.01378 0.01160 0.46The relationship between the test order shown in Figure4 and the corresponding proximity is obtained from Table 4.Figure 4
Cold roll-beating spline test order and the corresponding proximity.As shown in Figure4, the corresponding proximity of test order 4 is the best of all the proximities. As shown in Table 2, for test order 4, the corresponding roller speed is 1428 r/min and the feed rate is 0.206 mm/r (42 mm/min).


## 4.6. Verification of the Multiobjective Decision for the Cold Roll-Beating Spline Surface Performance

The grey correlation algorithm was used to calculate the grey correlative degree of the surface roughness, residual stress, and surface hardening degree of the surface performance, based on the order of the test and the grey correlation degree of the size, as shown in Figure5.Figure 5
Cold roll-beating spline test order and the corresponding grey correlation degree.Figure5 shows that test order 4 has a large grey correlation degree, as indicated in Table 2. For test order 4, the corresponding roller speed is 1428 r/min, and the feed rate is 0.206 mm/r (42 mm/min).Using the entropy weight ideal point method, an improved proximity is obtained for test order 4, and the grey correlation algorithm verifies that test order 4 has a large grey correlation degree, allowing better determination of the cold roll-beating spline processing parameters.


## 5. Conclusion

( 1 ) Based on the cold roll-beating spline test results, a multiobjective decision algorithm for entropy weight ideal points is constructed. This algorithm is applied to a physical cold roll-beating spline test, and the proximity of the test order to the ideal solution is obtained.( 2 ) The proximity of the multiobjective decision algorithm with entropy weight ideal points and the processing parameters of the cold roll-beating spline surface performance are optimized.



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*Source: 5048387-2018-03-04.xml*