Dai et al 2021a - Revision history

Shinsm2021 at 05:37, 4 May 2022

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Rimni at 14:40, 14 December 2021

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Scipediacontent at 08:37, 10 December 2021

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Rimni at 09:10, 8 October 2021

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Rimni at 09:08, 8 October 2021

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Rimni at 08:52, 8 October 2021

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Rimni at 08:51, 8 October 2021

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Rimni: /* 4. Numerical examples */

2021-10-08T08:36:08Z

‎4. Numerical examples

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-Accordingly, if player 2 is leader and player 1 is the follower, the Stackelberg equilibrium solution can be obtained in the objective function of player 2 if:
+Accordingly, if player 2 is the leader and player 1 is the follower, the Stackelberg equilibrium solution can be obtained in the objective function of player 2 if:
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-| style="text-align: center;" | <math>\mathrm{{\tilde{\boldsymbol\mathrm{x}}}_{\sigma }=}\,\mathrm{min}_{{\boldsymbol\mathrm{x}}_{\sigma }\in X}\,{\hat{\sigma }}^{2}({\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma })=\frac{\partial {\hat{\sigma }}^{2}({\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma })}{\partial {\boldsymbol\mathrm{x}}_{\sigma }}</math>
+| style="text-align: center;" | <math>\mathrm{{\tilde{\boldsymbol\mathrm{x}}}_{\sigma }=}\,\mathrm{argmin}_{{\boldsymbol\mathrm{x}}_{\sigma }\in X}\,{\hat{\sigma }}^{2}({\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma })=\frac{\partial {\hat{\sigma }}^{2}({\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma })}{\partial {\boldsymbol\mathrm{x}}_{\sigma }}</math>
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 |}
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-| style="text-align: center;" | <math>\mathrm{{\tilde{\boldsymbol\mathrm{x}}}_{\mu }=}\,\mathrm{min}_{{\boldsymbol\mathrm{x}}_{\mu }\in X}\,{(\hat{\mu }\left( {\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma }\right) -\tau )}^{2}=\frac{\partial {(\hat{\mu }\left( {\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma }\right) -\tau )}^{2}}{\partial {\boldsymbol\mathrm{x}}_{\mu }}</math>
+| style="text-align: center;" | <math>\mathrm{{\tilde{\boldsymbol\mathrm{x}}}_{\mu }=}\,\mathrm{argmin}_{{\boldsymbol\mathrm{x}}_{\mu }\in X}\,{(\hat{\mu }\left( {\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma }\right) -\tau )}^{2}=\frac{\partial {(\hat{\mu }\left( {\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma }\right) -\tau )}^{2}}{\partial {\boldsymbol\mathrm{x}}_{\mu }}</math>
 |}
 |}
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 Comparing the non-dominated results of the bias-leader and variance-leader models in example 1 based on the MSE criterion shows that the variance-leader model provides a more efficient outcome, and it is more appropriate to optimize process variance in advance when the process bias is treated as a restriction. In addition, <math display="inline">{x}_{1}</math> and <math display="inline">{x}_{2}</math> are regarded as the controlled factors for process bias, and the solution set is <math display="inline">\left\{ {\left( \hat{\mu }\left( {\boldsymbol\mathrm{x}}^{\ast }\right) -\tau \right) }^{2},\, {\hat{\sigma }}^{2}\left( {\boldsymbol\mathrm{x}}^{\ast }\right) \right\} =</math><math>(0,\, 2\, 049.10)</math>.
-A comparison of the significant results of example 1 between the dual response approach in <span id='cite-_Ref76149715'></span>[[#_Ref76149715|[2]]] and the proposed SGRD model is exhibited in [[#tab-5|Table 5]]. As evidenced by the results, the proposed SGRD model is more efficient than the dual response model based on the same zero-bias condition. Since the dual response approach is a pure priority-based optimization method, the comparison also shows the importance of considering the correlations between controlled factors and optimization objectives. The contour plots for the process mean and standard deviation are shown in [[#img-4|Figure 4]] , where the Stackelberg equilibrium solutions of the bias-leader model and variance-leader model are denoted by dots. The solution obtained by usingthe  dual response approach is plotted as a star, and the most efficient solution provided by the proposed SGRD model is marked as a circle.
+A comparison of the significant results of example 1 between the dual response approach in <span id='cite-_Ref76149715'></span>[[#_Ref76149715|[2]]] and the proposed SGRD model is exhibited in [[#tab-5|Table 5]]. As evidenced by the results, the proposed SGRD model is more efficient than the dual response model based on the same zero-bias condition. Since the dual response approach is a pure priority-based optimization method, the comparison also shows the importance of considering the correlations between controlled factors and optimization objectives. The contour plots for the process mean and standard deviation are shown in [[#img-4|Figure 4]], where the Stackelberg equilibrium solutions of the bias-leader model and variance-leader model are denoted by dots. The solution obtained by usingthe  ual response approach is plotted as a star, and the most efficient solution provided by the proposed SGRD model is marked as a circle.
 <div class="center" style="font-size: 75%;">'''Table 5'''. Comparison of the optimization results (example 1) </div>

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 The feasible solution set <math display="inline">X=\left\{ \boldsymbol\mathrm{x}\in {R}^{3}:\, g\left( \boldsymbol\mathrm{x}\right) =\right. </math><math>\left. \sum _{m=1}^{3}{x}_{m}^{2}-3\leq 0,\, m=1,2,3\right\}.</math> The target coating thickness is specified by the customer and is set at 71.14. The goal of the bi-objective optimization problem is to minimize process bias and variability. Based on the same optimization progress in example 1, the proposed SGRD model can be decomposed into a bias-leader model and variance-leader model, which are stated below.
-====4.2.1. Bias-leader model====
+====4.2.1 Bias-leader model====
 In the bias-leader model, process variance is imagined as the optimization constraints while optimizing the process bias, and various combinations of the controlled variables have been taken into account. Eq. (7) of the proposed SGRD model is applied, and the experimental outcome is shown in [[#tab-8|Table 8]].
@@ Line 1,177: / Line 1,177: @@
 According to the MSE criterion, the solution set <math display="inline">\left\{ {\left( \hat{\mu }\left( {\boldsymbol\mathrm{x}}^{\ast }\right) -\tau \right) }^{2},\, {\hat{\sigma }}^{2}\left( {\boldsymbol\mathrm{x}}^{\ast }\right) \right\} =</math><math>\left( 1.2160,\, 5.7586\right)</math>  with input variable setting <math display="inline">({x}_{1}^{\ast },\, {x}_{2}^{\ast },\, {x}_{3}^{\ast })=(-0.0778,\, 0.2155,\, -</math><math>0.2813)</math> has been identified as the non-dominated result in this model, where the MSE value is 6.9746. Moreover, controlled factors <math display="inline">{x}_{1}</math> and <math display="inline">{x}_{3}</math> are assumed to be significantly correlated with the game follower, which is process variance.
-====4.2.2. Variance-leader model====
+====4.2.2 Variance-leader model====
 Using Eq. (8) from the proposed SGRD model yields the results listed in [[#tab-9|Table 9]]. The non-dominated solution set of the variance-leader model is <math display="inline">\left\{ {\left( \hat{\mu }\left( {\boldsymbol\mathrm{x}}^{\ast }\right) -\tau \right) }^{2},\, {\hat{\sigma }}^{2}\left( {\boldsymbol\mathrm{x}}^{\ast }\right) \right\} =</math><math>(1.2414,\, 5.7351),</math> with a minimum MSE value of 6.9765. The corresponding control variable setting is <math display="inline">({x}_{1}^{\ast },\, {x}_{2}^{\ast },\, {x}_{3}^{\ast })=(-0.0158,\, 0.3130,\, -</math><math>0.3021)</math>, and the input factor <math display="inline">{x}_{2}</math> is regarded as controlled by the process bias.
@@ Line 1,245: / Line 1,245: @@
 |}
-====4.2.3. Results and discussion====
+====4.2.3 Results and discussion====
 A comparison of the non-dominated results of the bias-leader and variance-leader models in example 2 based on the MSE criterion shows that the result of the bias-leader model is slightly better than that of the variance-leader model.

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 === 4.1 Example 1 ===
 A printing machine example is discussed in <span id='cite-_Ref76149715'></span>[[#_Ref76149715|[2]]]. The authors investigated how the printer's ability (<math display="inline">y</math>) to apply colored inks to package labels is affected by speed (<math display="inline">{x}_{1}</math>), pressure (<math display="inline">{x}_{2}</math>), and distance (<math display="inline">{x}_{3}</math>). This is a three-level factorial experimental design with three runs (<math display="inline">{r}_{i}</math>) at each design point. Detailed experimental data is shown in [[#tab-2|Table 2]], where <math display="inline">\overline{y}</math> and <math display="inline">s</math> represent the process mean and process standard deviation, respectively.
 <div class="center" style="font-size: 75%;">'''Table 2'''. Printing study data [[#_Ref76149715|[2]]]</div>

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 === 3.4 Proposed SGRD model ===
-The integration procedure of the proposed bi-objective SGRD optimization problem from a Stackelberg game model is shown in Fig. 3. The process bias and variance can be identified as the two game players of the proposed SGRD model. Then, the objective functions introduced in the Stackelberg leader-follower model, <math display="inline">{f}_{1}\left( x,y\right)</math>  and <math display="inline">{f}_{2}\left( x,y\right)</math> , can be replaced by the estimated functions of process bias <math display="inline">{(\hat{\mu }\left( \boldsymbol\mathrm{x}\right) -\tau )}^{2}</math> and variance <math display="inline">{\hat{\sigma }}^{2}(\boldsymbol\mathrm{x})</math>, respectively. The optimization objectives of the bi-objective optimization problem <math display="inline">\left\{ {(\hat{\mu }\left( \boldsymbol\mathrm{x}\right) -\tau )}^{2},\, {\hat{\sigma }}^{2}(\boldsymbol\mathrm{x})\right\}</math>  and the input factors <math display="inline">\boldsymbol\mathrm{x}</math> are separately controlled by the process bias and variance. We can define the input factors that are controlled by the process bias are denoted by <math display="inline">\, {\boldsymbol\mathrm{x}}_{\mu }</math>, and <math display="inline">\, {\boldsymbol\mathrm{x}}_{\mu }\subseteq \boldsymbol\mathrm{x}</math>. Correspondingly,''' '''as the complement of <math display="inline">{\boldsymbol\mathrm{x}}_{\mu }</math>, the input factors that are controlled by the process variance are denoted by <math display="inline">{\boldsymbol\mathrm{\, x}}_{\sigma }</math>,where <math display="inline">{\boldsymbol\mathrm{x}}_{\sigma }=</math><math>{{\boldsymbol\mathrm{x}}_{\mu }}^{c}\subseteq \boldsymbol\mathrm{x}</math>. Similar to the Stackelberg game model, the response functions of process bias and variance are depended on the controlled factors <math display="inline">{\boldsymbol\mathrm{x}}_{\mu }</math> and <math display="inline">{\boldsymbol\mathrm{x}}_{\sigma }</math>. Thus, the objective functions for the process bias and variance can be written as <math display="inline">{(\hat{\mu }\left( {\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma }\right) -\tau )}^{2}</math> and <math display="inline">{\hat{\sigma }}^{2}({\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma })</math>, respectively. Based on the Stackelberg game rules, the target of process bias is to minimize <math display="inline">{(\hat{\mu }\left( {\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma }\right) -\tau )}^{2}</math> and considers only factor <math display="inline">{\boldsymbol\mathrm{x}}_{\mu }</math>. Accordingly, the target of process variance is to minimize <math display="inline">{\hat{\sigma }}^{2}({\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma })</math> and considers only factor <math display="inline">{\boldsymbol\mathrm{x}}_{\sigma }</math>.
+The integration procedure of the proposed bi-objective SGRD optimization problem from a Stackelberg game model is shown in [[#img-3|Figure 3]]. The process bias and variance can be identified as the two game players of the proposed SGRD model. Then, the objective functions introduced in the Stackelberg leader-follower model, <math display="inline">{f}_{1}\left( x,y\right)</math>  and <math display="inline">{f}_{2}\left( x,y\right)</math> , can be replaced by the estimated functions of process bias <math display="inline">{(\hat{\mu }\left( \boldsymbol\mathrm{x}\right) -\tau )}^{2}</math> and variance <math display="inline">{\hat{\sigma }}^{2}(\boldsymbol\mathrm{x})</math>, respectively. The optimization objectives of the bi-objective optimization problem <math display="inline">\left\{ {(\hat{\mu }\left( \boldsymbol\mathrm{x}\right) -\tau )}^{2},\, {\hat{\sigma }}^{2}(\boldsymbol\mathrm{x})\right\}</math>  and the input factors <math display="inline">\boldsymbol\mathrm{x}</math> are separately controlled by the process bias and variance. We can define the input factors that are controlled by the process bias are denoted by <math display="inline">\, {\boldsymbol\mathrm{x}}_{\mu }</math>, and <math display="inline">\, {\boldsymbol\mathrm{x}}_{\mu }\subseteq \boldsymbol\mathrm{x}</math>. Correspondingly,''' '''as the complement of <math display="inline">{\boldsymbol\mathrm{x}}_{\mu }</math>, the input factors that are controlled by the process variance are denoted by <math display="inline">{\boldsymbol\mathrm{\, x}}_{\sigma }</math>,where <math display="inline">{\boldsymbol\mathrm{x}}_{\sigma }=</math><math>{{\boldsymbol\mathrm{x}}_{\mu }}^{c}\subseteq \boldsymbol\mathrm{x}</math>. Similar to the Stackelberg game model, the response functions of process bias and variance are depended on the controlled factors <math display="inline">{\boldsymbol\mathrm{x}}_{\mu }</math> and <math display="inline">{\boldsymbol\mathrm{x}}_{\sigma }</math>. Thus, the objective functions for the process bias and variance can be written as <math display="inline">{(\hat{\mu }\left( {\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma }\right) -\tau )}^{2}</math> and <math display="inline">{\hat{\sigma }}^{2}({\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma })</math>, respectively. Based on the Stackelberg game rules, the target of process bias is to minimize <math display="inline">{(\hat{\mu }\left( {\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma }\right) -\tau )}^{2}</math> and considers only factor <math display="inline">{\boldsymbol\mathrm{x}}_{\mu }</math>. Accordingly, the target of process variance is to minimize <math display="inline">{\hat{\sigma }}^{2}({\boldsymbol\mathrm{x}}_{\mu },{\boldsymbol\mathrm{x}}_{\sigma })</math> and considers only factor <math display="inline">{\boldsymbol\mathrm{x}}_{\sigma }</math>.
 <div id='img-3'></div>

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 ====4.2.2. Variance-leader model====
-Using Eq. (8) from the proposed SGRD model yields the results listed in Table 9. The non-dominated solution set of the variance-leader model is <math display="inline">\left\{ {\left( \hat{\mu }\left( {\boldsymbol\mathrm{x}}^{\ast }\right) -\tau \right) }^{2},\, {\hat{\sigma }}^{2}\left( {\boldsymbol\mathrm{x}}^{\ast }\right) \right\} =</math><math>(1.2414,\, 5.7351),</math> with a minimum MSE value of 6.9765. The corresponding control variable setting is <math display="inline">({x}_{1}^{\ast },\, {x}_{2}^{\ast },\, {x}_{3}^{\ast })=(-0.0158,\, 0.3130,\, -</math><math>0.3021)</math>, and the input factor <math display="inline">{x}_{2}</math> is regarded as controlled by the process bias.
+Using Eq. (8) from the proposed SGRD model yields the results listed in [[#tab-9|Table 9]]. The non-dominated solution set of the variance-leader model is <math display="inline">\left\{ {\left( \hat{\mu }\left( {\boldsymbol\mathrm{x}}^{\ast }\right) -\tau \right) }^{2},\, {\hat{\sigma }}^{2}\left( {\boldsymbol\mathrm{x}}^{\ast }\right) \right\} =</math><math>(1.2414,\, 5.7351),</math> with a minimum MSE value of 6.9765. The corresponding control variable setting is <math display="inline">({x}_{1}^{\ast },\, {x}_{2}^{\ast },\, {x}_{3}^{\ast })=(-0.0158,\, 0.3130,\, -</math><math>0.3021)</math>, and the input factor <math display="inline">{x}_{2}</math> is regarded as controlled by the process bias.
 <div class="center" style="font-size: 75%;">'''Table 9.''' Variance-leader Stackelberg game results (example 2)</div>
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 | style="text-align: center;" |10.4374
 |}
 ====4.2.3. Results and discussion====

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 | colspan="1" style="padding:10px;"| '''Figure 5'''. Contour plots for the process mean and standard deviation (example 2)
 |}
 ==5. Conclusion==

@@ Line 1,109: / Line 1,109: @@
 In the bias-leader model, process variance is imagined as the optimization constraints while optimizing the process bias, and various combinations of the controlled variables have been taken into account. Eq. (7) of the proposed SGRD model is applied, and the experimental outcome is shown in [[#tab-8|Table 8]].
-<div class="center" style="font-size: 75%;">'''Table 8'''. Bias-leader Stackelberg game results (example 2)
+<div class="center" style="font-size: 75%;">'''Table 8'''. Bias-leader Stackelberg game results (example 2)</div>
 <div id='tab-8'></div>