Zhang et al 2023i - Revision history

D.S.A at 00:01, 6 December 2023

2023-12-06T00:01:32Z

D.S.A at 00:00, 6 December 2023

2023-12-06T00:00:55Z

D.S.A at 04:03, 1 November 2023

2023-11-01T04:03:53Z

Rimni at 12:03, 27 October 2023

2023-10-27T12:03:21Z

Rimni: /* 2.3 Maximizing EI */

2023-10-26T14:22:06Z

‎2.3 Maximizing EI

Rimni at 14:20, 26 October 2023

2023-10-26T14:20:30Z

Rimni: /* 2.1 KRG model */

2023-10-26T14:14:50Z

‎2.1 KRG model

Rimni at 14:10, 26 October 2023

2023-10-26T14:10:56Z

Rimni at 14:08, 26 October 2023

2023-10-26T14:08:45Z

Rimni at 14:06, 26 October 2023

2023-10-26T14:06:48Z

@@ Line 4: / Line 4: @@
 ==Adaptive ensemble surrogate-based optimization and analysis of forklift pallet racks==
-<big>Wei Zhang<sup>1</sup>, Yujun Lu<sup>1</sup></big>, Yaze Mei<sup>1</sup>, Liye Lv<sup>1</sup>
+<big>Wei Zhang<sup>1</sup>, Yujun Lu<sup>1</sup></big>, Yaze Mei<sup>1</sup>, Liye Lv<sup>1</sup>, Baochang Zhu<sup>1</sup>
 School of Mechanical Engineering, Zhejiang Sci-Tech University, Hangzhou, Zhejiang, China, 310018

@@ Line 696: / Line 696: @@
 ==Acknowledgment==
-The research was supported by the Longgang Institute of Zhejiang Sci-Tech University(LGYJY2021004) and Zhejiang Province Science and Technology Plan Project (2023C01174).
+The research was supported by Zhejiang University of Technology Project(11132932612108), the Longgang Institute of Zhejiang Sci-Tech University(LGYJY2021004), and Zhejiang Province Science and Technology Plan Project (2023C01174).
 ==References==

@@ Line 696: / Line 696: @@
 ==Acknowledgment==
-The research was supported by the Longgang Institute of Zhejiang Sci-Tech University(LGYJY2021004).
+The research was supported by the Longgang Institute of Zhejiang Sci-Tech University(LGYJY2021004) and Zhejiang Province Science and Technology Plan Project (2023C01174).
 ==References==

@@ Line 121: / Line 121: @@
 {| style="text-align: center;margin:auto;width: 100%;"
 |-
-| <math>{\omega }_{i}=\frac{{P}_{i}}{\sum _{j=1}^{M}{P}_{j}}</math>
+| <math>{\omega }_{i}=\frac{{P}_{i}}{\displaystyle\sum _{j=1}^{M}{P}_{j}}</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(6)

@@ Line 137: / Line 137: @@
 {| style="text-align: center;margin:auto;width: 100%;"
 |-
-| <math>E\left[ I\left( x\right) \right] =\left( {y}_{\min}-\hat{y}(x) \right) \left[ \frac{1}{2}+ \frac{1}{2}\mathrm{erf}\,\left( \frac{{y}_{\min}-\hat{y}(x) }{\hat{s}\sqrt{2}}\right) \right] +</math><math>\hat{s}\frac{1}{\sqrt{2\pi }}\exp\left[ \frac{-({{y}_{\min}-\hat{y}(x))}^{2}}{2\hat{s}}\right]</math>
+| <math>E\left[ I(x) \right] =\left( {y}_{\min}-\hat{y}(x) \right) \left[ \frac{1}{2}+ \frac{1}{2}\mathrm{erf}\,\left( \frac{{y}_{\min}-\hat{y}(x) }{\hat{s}\sqrt{2}}\right) \right] +</math><math>\hat{s}\frac{1}{\sqrt{2\pi }}\exp\left[ \frac{-({{y}_{\min}-\hat{y}(x))}^{2}}{2\hat{s}}\right]</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(7)
 |}
-It is observed from equations <math display="inline">P\left[ I\left( x\right) \right]</math> and  <math display="inline">E\left[ I(x) \right]</math>  that the expressions for both PoI and EI strategies contain global terms (<math display="inline">{s}^{2}</math>) and local terms (<math display="inline">I=</math><math>{y}_{\min}-Y(x)</math>), indicating that both approaches can balance global exploration and local exploitation to some extent.
+It is observed from equations <math display="inline">P\left[ I(x) \right]</math> and  <math display="inline">E\left[ I(x) \right]</math>  that the expressions for both PoI and EI strategies contain global terms (<math display="inline">{s}^{2}</math>) and local terms (<math display="inline">I=</math><math>{y}_{\min}-Y(x)</math>), indicating that both approaches can balance global exploration and local exploitation to some extent.
 ===2.4 Performance criterion===

@@ Line 41: / Line 41: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>y\left( \boldsymbol{x}\right) =f\left( \boldsymbol{x}\right) +z\left( \boldsymbol{x}\right)</math>
+| style="text-align: center;" | <math>y( \boldsymbol{x}) =f( \boldsymbol{x}) +z( \boldsymbol{x})</math>
 |}
 |  style="text-align: center;vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(1)
@@ Line 55: / Line 55: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>R\left( {\boldsymbol{x}}^{i},{\boldsymbol{x}}^{j}\right) =\mathrm{exp}\,\left[ -\sum _{k=1}^{{n}_{dv}}{\theta }_{k}{\left| {x}_{k}^{i}-{x}_{k}^{j}\right| }^{{P}_{k}}\right]</math>
+| style="text-align: center;" | <math>R( {\boldsymbol{x}}^{i},{\boldsymbol{x}}^{j}) =\mathrm{exp}\,\left[ -\sum _{k=1}^{{n}_{dv}}{\theta }_{k}{\left| {x}_{k}^{i}-{x}_{k}^{j}\right| }^{{P}_{k}}\right]</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(2)
@@ Line 69: / Line 69: @@
 {| style="text-align: center; margin:auto;width: 100%;"
 |-
-| style="text-align: center;" | <math>\hat{y}\left( x\right) =\hat{\beta }+\boldsymbol{r}\left( x\right) {\boldsymbol{R}}^{-1}\left( y-\boldsymbol{f}\hat{\beta }\right)</math>
+| style="text-align: center;" | <math>\hat{y}(x) =\hat{\beta }+\boldsymbol{r}(x) {\boldsymbol{R}}^{-1}( y-\boldsymbol{f}\hat{\beta })</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(3)
@@ Line 137: / Line 137: @@
 {| style="text-align: center;margin:auto;width: 100%;"
 |-
-| <math>E\left[ I\left( x\right) \right] =\left( {y}_{\min}-\hat{y}\left( x\right) \right) \left[ \frac{1}{2}+ \frac{1}{2}\mathrm{erf}\,\left( \frac{{y}_{\min}-\hat{y}\left( x\right) }{\hat{s}\sqrt{2}}\right) \right] +</math><math>\hat{s}\frac{1}{\sqrt{2\pi }}\exp\left[ \frac{-({{y}_{\min}-\hat{y}(x))}^{2}}{2\hat{s}}\right]</math>
+| <math>E\left[ I\left( x\right) \right] =\left( {y}_{\min}-\hat{y}(x) \right) \left[ \frac{1}{2}+ \frac{1}{2}\mathrm{erf}\,\left( \frac{{y}_{\min}-\hat{y}(x) }{\hat{s}\sqrt{2}}\right) \right] +</math><math>\hat{s}\frac{1}{\sqrt{2\pi }}\exp\left[ \frac{-({{y}_{\min}-\hat{y}(x))}^{2}}{2\hat{s}}\right]</math>
 |}
 |  style="text-align: center;width: 5px;text-align: right;white-space: nowrap;"|(7)
 |}
-It is observed from equations <math display="inline">P\left[ I\left( x\right) \right]</math> and  <math display="inline">E\left[ I\left( x\right) \right]</math>  that the expressions for both PoI and EI strategies contain global terms (<math display="inline">{s}^{2}</math>) and local terms (<math display="inline">I=</math><math>{y}_{\min}-Y\left( x\right)</math>), indicating that both approaches can balance global exploration and local exploitation to some extent.
+It is observed from equations <math display="inline">P\left[ I\left( x\right) \right]</math> and  <math display="inline">E\left[ I(x) \right]</math>  that the expressions for both PoI and EI strategies contain global terms (<math display="inline">{s}^{2}</math>) and local terms (<math display="inline">I=</math><math>{y}_{\min}-Y(x)</math>), indicating that both approaches can balance global exploration and local exploitation to some extent.
 ===2.4 Performance criterion===
-In this paper, we use two criteria to evaluate the accuracy of the model. That is, the coefficient of determination <math display="inline">\left( {R}^{2}\right)</math> is used as a global evaluation metric; the normalized max absolute error (NMAE) is used as a local evaluation metric to evaluate the global and local performance of different infilling strategies, respectively
+In this paper, we use two criteria to evaluate the accuracy of the model. That is, the coefficient of determination <math display="inline">( {R}^{2})</math> is used as a global evaluation metric; the normalized max absolute error (NMAE) is used as a local evaluation metric to evaluate the global and local performance of different infilling strategies, respectively
 {| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"

@@ Line 46: / Line 46: @@
 |}
-where  <math display="inline">y\left( \boldsymbol{x}\right)</math>  is the response function for the region of interest,  <math display="inline">f\left( \boldsymbol{x}\right)</math>  is a polynomial function used as an approximation to the underlying functional model, and <math display="inline">z\left( \boldsymbol{x}\right)</math>  denotes a stochastic process with uncertainty in the mean of <math display="inline">f\left( \boldsymbol{x}\right)</math>. The KRG model is an interpolation method through each sample point.
+where  <math display="inline">y( \boldsymbol{x})</math>  is the response function for the region of interest,  <math display="inline">f( \boldsymbol{x})</math>  is a polynomial function used as an approximation to the underlying functional model, and <math display="inline">z( \boldsymbol{x})</math>  denotes a stochastic process with uncertainty in the mean of <math display="inline">f( \boldsymbol{x})</math>. The KRG model is an interpolation method through each sample point.
 In this paper, we use the Gaussian correlation function：
@@ Line 60: / Line 60: @@
 |}
-where <math display="inline">{n}_{dv}</math> is the number of design variables, <math display="inline">\theta</math>  and  <math display="inline">P</math> are parameters that vary flexibly with the variables, and  <math display="inline">\left| {x}_{k}^{i}-{x}_{k}^{j}\right|</math> is the distance between  <math display="inline">{\boldsymbol{x}}^{i}</math> and  <math display="inline">{\boldsymbol{x}}^{j}</math> concerning the k''-''th variable.
+where <math display="inline">{n}_{dv}</math> is the number of design variables, <math display="inline">\theta</math>  and  <math display="inline">P</math> are parameters that vary flexibly with the variables, and  <math display="inline">\left| {x}_{k}^{i}-{x}_{k}^{j}\right|</math> is the distance between  <math display="inline">{\boldsymbol{x}}^{i}</math> and  <math display="inline">{\boldsymbol{x}}^{j}</math> concerning the <math display="inline">k</math>-th variable.
-The predicted value of  <math display="inline">\hat{y}\left( \boldsymbol{x}\right)</math>  at test point <math> x </math> is formulated as follows：
+The predicted value of  <math display="inline">\hat{y}(\boldsymbol{x})</math>  at test point <math> x </math> is formulated as follows：
 {| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
@@ Line 74: / Line 74: @@
 |}
-where  <math display="inline">\boldsymbol{y}</math> is a column vector of sample response values, and when  <math display="inline">f\left( \boldsymbol{x}\right)</math>  is a constant vector,  <math display="inline">\boldsymbol{f}</math> is a column vector of length <math> n</math> with all elements of 1. <math display="inline">\boldsymbol{r}\left( x\right)</math>  is the correlation vector between the test point<math> x </math> and the sample points.
+where  <math display="inline">\boldsymbol{y}</math> is a column vector of sample response values, and when  <math display="inline">f( \boldsymbol{x})</math>  is a constant vector,  <math display="inline">\boldsymbol{f}</math> is a column vector of length <math> n</math> with all elements of 1. <math display="inline">\boldsymbol{r}(x)</math>  is the correlation vector between the test point<math> x </math> and the sample points.
 ===2.2 Ensemble surrogate models===

@@ Line 110: / Line 110: @@
 |}
-where  <math display="inline">{y}_{basej}</math> is the prediction value of the baseline model at the j-th sample point, Pi is the probability coefficient associated with the i-th individual surrogate model, and  <math display="inline">{s}_{j}^{2}</math> is the local prediction esti-mation at the j-th sample point.
+where  <math display="inline">{y}_{basej}</math> is the prediction value of the baseline model at the <math display="inline">j</math>-th sample point, <math display="inline">P_i</math> is the probability coefficient associated with the <math display="inline">i</math>-th individual surrogate model, and  <math display="inline">{s}_{j}^{2}</math> is the local prediction estimation at the <math display="inline">j</math>-th sample point.
 '''Step 3. Local weight determination'''
@@ Line 168: / Line 168: @@
 |}
-where  <math display="inline">{y}_{i}</math> denotes the true response at the i-th test point, and the predicted value of the model built by the surrogate model approach at the i-th test point is  <math display="inline">{\hat{y}}_{i}</math>,   and <math display="inline">\overline{y}</math> is the mean value of the true response at all test points, ntest denotes the number of test points.
+where  <math display="inline">{y}_{i}</math> denotes the true response at the <math display="inline">i</math>-th test point, and the predicted value of the model built by the surrogate model approach at the <math display="inline">i</math>-th test point is  <math display="inline">{\hat{y}}_{i}</math>,   and <math display="inline">\overline{y}</math> is the mean value of the true response at all test points, ntest denotes the number of test points.
 Usually, <math display="inline">{R}^{2}</math> ranges from 0 to 1, and a larger d or a smaller NMAE means more accurate surrogates. When  <math display="inline">{R}^{2}</math> is negative, it represents that the constructed surrogate model is trending opposite to the true model.

@@ Line 629: / Line 629: @@
 |}
-where <math display="inline">{x}_{i}</math> denotes the i-th variable and <math display="inline">{x}_{j}</math> denotes the <math display="inline">j</math>-th variable.
+where <math display="inline">{x}_{i}</math> denotes the <math display="inline">i</math>-th variable and <math display="inline">{x}_{j}</math> denotes the <math display="inline">j</math>-th variable.
 The total standard deviation and partial standard deviation of <math display="inline">f(x)</math> are, respectively:
@@ Line 667: / Line 667: @@
 |}
-where <math display="inline">{S}_{i}</math> denotes the first-order sensitivity index of the i-th variable. The global sensitivity index  <math display="inline">{S}_{ti}</math> of the <math display="inline">i</math>-th variable is then the sum of the sensitivity indices of each order of the i-th variable.
+where <math display="inline">{S}_{i}</math> denotes the first-order sensitivity index of the <math display="inline">i</math>-th variable. The global sensitivity index  <math display="inline">{S}_{ti}</math> of the <math display="inline">i</math>-th variable is then the sum of the sensitivity indices of each order of the <math display="inline">i</math>-th variable.
 Based on the E-AHF model, the GSA analysis of the pallet rack is carried out, and the results are shown in [[#img-10|Figure 10]]. It can be clearly seen that the channel height <math display="inline">{x}_{1}</math> has the greatest influence on the shape variables of the pallet rack, and both the first-order sensitivity index and the global sensitivity index are much higher than the remaining four design variables; whereas, the one that has a greater influence on the stress of the pallet rack is the channel flange thickness  <math display="inline">{x}_{4}</math>, and its sensitivity index is also much higher than that of the other four variables; whereas, for the pallet rack mass, it can be seen that, except for the web thickness  <math display="inline">{x}_{3}</math>, the rest of the design variables all have a greater influence. In summary, in the design of a pallet rack, the highest priority should be given to the height of channel steel <math display="inline">{x}_{1}</math> which has the greatest influence on its shape variables, to ensure the overall safety and pressure resistance of the lifting platform.

@@ Line 510: / Line 510: @@
 Based on the structural values derived from the optimization objective of minimizing deformation, in order to check the accuracy of the optimization results (incorporating stress, deformation and mass) predicted by the two surrogate models for the values calculated by the optimization algorithms, the optimization results were brought into the simulation model to obtain the real results and the errors were obtained by comparing them with the predicted values of the models, which are shown in [[#tab-5|Tables 5]], [[#tab-6|6]] and [[#tab-7|7]], respectively.
-<<div class="center" style="font-size: 75%;">'''Table 5'''. Deformation results of before and after optimization</div>
+<div class="center" style="font-size: 75%;">'''Table 5'''. Deformation results of before and after optimization</div>
 <div id='tab-5'></div>
@@ Line 538: / Line 538: @@
-<<div class="center" style="font-size: 75%;">'''Table 6'''. Stress results of before and after optimization</div>
+<div class="center" style="font-size: 75%;">'''Table 6'''. Stress results of before and after optimization</div>
 <div id='tab-6'></div>
@@ Line 566: / Line 566: @@
-<<div class="center" style="font-size: 75%;">'''Table 7'''. Quality results of before and after optimization</div>
+<div class="center" style="font-size: 75%;">'''Table 7'''. Quality results of before and after optimization</div>
 <div id='tab-7'></div>