Tang et al 2022a - Revision history

Rimni: /* References */

2022-06-20T13:41:47Z

‎References

Rimni at 13:34, 20 June 2022

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Rimni at 12:40, 17 June 2022

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Rimni: /* 5.2 Sensitivity analysis for numerical example 1 */

2022-06-17T11:16:49Z

‎5.2 Sensitivity analysis for numerical example 1

Rimni at 11:14, 17 June 2022

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Rimni: /* 5.1 Numerical example 1 */

2022-06-17T10:17:02Z

‎5.1 Numerical example 1

Rimni at 10:14, 17 June 2022

2022-06-17T10:14:16Z

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@@ Line 1,405: / Line 1,405: @@
 [55] Mandal W.A. Weighted Tchebycheff optimization technique under uncertainty. Annals of Data Science, 8:709–731, 2021.
-[56] Dächert K, Gorski J, Klamroth K. An augmented weighted Tchebycheff method with adaptively chosen parameters for discrete bicriteria optimization problems. Computers & Operations Research, 39(12): 2929-2943, 2012.
+[56] Dächert K., Gorski J., Klamroth K. An augmented weighted Tchebycheff method with adaptively chosen parameters for discrete bicriteria optimization problems. Computers & Operations Research, 39(12):2929-2943, 2012.
-[57] Steuer R E, Choo E U. An interactive weighted Tchebycheff procedure for multiple objective programming. Mathematical programming, 26(3): 326-344, 1983.
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-[59] Myerson R B. Game Theory: Analysis of Conflict. Harvard University Press, Cambridge, MA. London England, 1991.
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-[60] Thomson W. In: Handbook of game theory with economic applications. Cooperative models of Bargaining. 2: 1237-1284, 1994.
+[60] Thomson W. Chapter 35: Cooperative models of Bargaining. In: Handbook of Game Theory with Economic Applications, 2:1237-1284, 1994.

@@ Line 1,363: / Line 1,363: @@
 [34] Barough A.S., Shoubi M.V., Skardi M.J.E. Application of game theory approach in solving the construction project conflicts. Procedia-Social and Behavioral Sciences, 58:1586-1593, 2012.
-[35] Gale D., Kuhn H.W., Tucker A.W. Linear programming and the theory of game. Activity analysis of production and allocation, 13: 317-335, 1951.
+[35] Gale D., Kuhn H.W., Tucker A.W. Linear programming and the theory of game. Activity analysis of production and allocation, 13:317-335, 1951.
-[36] Mangasarian O L, Stone H. Two-person nonzero-sum games and quadratic programming. Journal of mathematical analysis and applications, 9(3): 348-355, 1964.
+[36] Mangasarian O.L., Stone H. Two-person nonzero-sum games and quadratic programming. Journal of Mathematical Analysis and Applications, 9(3):348-355, 1964.
-[37] Leboucher C, Shin H S, Siarry P, et al. Convergence proof of an enhanced particle swarm optimization method integrated with evolutionary game theory. Information Sciences, 346: 389-411, 2016.
+[37] Leboucher C., Shin H.S., Siarry P., et al. Convergence proof of an enhanced particle swarm optimization method integrated with evolutionary game theory. Information Sciences, 346:389-411, 2016.
-[38] Annamdas K K, Rao S S. Multi-objective optimization of engineering systems using game theory and particle swarm optimization. Engineering optimization, 41(8): 737-752, 2009.
+[38] Annamdas K.K., Rao S.S. Multi-objective optimization of engineering systems using game theory and particle swarm optimization. Engineering Optimization, 41(8):737-752, 2009.
-[39] Zamarripa, M. A., Aguirre, A. M., Méndez, C. A., & Espuña, A.  Mathematical programming and game theory optimization-based tool for supply chain planning in cooperative/competitive environments. Chemical Engineering Research and Design, 91(8): 1588-1600, 2013.
+[39] Zamarripa M.A., Aguirre A.M., Méndez C.A.,  Espuña A.  Mathematical programming and game theory optimization-based tool for supply chain planning in cooperative/competitive environments. Chemical Engineering Research and Design, 91(8):1588-1600, 2013.
-[40] Dai, L., Tang, M., & Shin, S. Stackelberg game approach to a bi-objective robust design optimization. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 37(4), 2021.
+[40] Dai L., Tang M., Shin S. Stackelberg game approach to a bi-objective robust design optimization. Revista Internacional de Métodos Numéricos para Cálculo y Diseño en Ingeniería, 37(4), 2021.
-[41] Matejaš J, Perić T. A new iterative method for solving multiobjective linear programming problem. Applied mathematics and computation, 243: 746-754, 2014.
+[41] Matejaš J., Perić T. A new iterative method for solving multiobjective linear programming problem. Applied Mathematics and Computation, 243:746-754, 2014.
 [43] Muthoo A. Bargaining theory with applications. Cambridge University Press, 1999.
-[44] Goodpaster G. Rational decision-making in problem-solving negotiation: Compromise, interest-valuation, and cognitive error. Ohio St. J. on Disp. Resol. 8: 299, 1992.
+[44] Goodpaster G. Rational decision-making in problem-solving negotiation: Compromise, interest-valuation, and cognitive error. Ohio St. J. on Disp. Resol., 8:299-360, 1992.
-[45] Nash, J. F. The Bargaining Problem.  Econometrica, 18(2):155-162, 1950.
+[45] Nash J.F. The bargaining problem.  Econometrica, 18(2):155-162, 1950.
-[46] Nash, J. F.  Two-Person Cooperative Games. Econometrica, 21(1):128-140, 1953.
+[46] Nash J.F.  Two-person cooperative games. Econometrica, 21(1):128-140, 1953.
-[47] Kalai E, Smorodinsky M. Other solutions to Nash's bargaining problem.  Econometrica: Journal of the Econometric Society, 513-518, 1975.
+[47] Kalai E., Smorodinsky M. Other solutions to Nash's bargaining problem.  Econometrica: Journal of the Econometric Society, 43(3):513-518, 1975.
-[48] Rubinstein A.  Perfect equilibrium in a bargaining model.  Econometrica: Journal of the Econometric Society, 97-109, 1982.
+[48] Rubinstein A.  Perfect equilibrium in a bargaining model.  Econometrica: Journal of the Econometric Society, 50(1):97-109, 1982.
-[49] Köksoy, O. A nonlinear programming solution to robust multi-response quality problem. Applied mathematics and computation, 196(2), 603-612, 2008.
+[49] Köksoy O. A nonlinear programming solution to robust multi-response quality problem. Applied Mathematics and Computation, 196(2):603-612, 2008.
-[50] Goethals, P. L., & Cho, B. R. Extending the desirability function to account for variability measures in univariate and multivariate response experiments. Computers & Industrial Engineering, 62(2), 457-468, 2012.
+[50] Goethals P.L., Cho B.R. Extending the desirability function to account for variability measures in univariate and multivariate response experiments. Computers & Industrial Engineering, 62(2):457-468, 2012.
-[51] Wu, F. C., & Chyu, C. C.  Optimization of robust design for multiple quality characteristics. International Journal of Production Research, 42(2), 337-354, 2004.
+[51] Wu F.C., Chyu C.C.  Optimization of robust design for multiple quality characteristics. International Journal of Production Research, 42(2):337-354, 2004.
-[52] Shin, S., & Cho, B. R. Bias-specified robust design optimization and its analytical solutions. Computers & Industrial Engineering, 48(1), 129-140, 2005.
+[52] Shin S.,  Cho B.R. Bias-specified robust design optimization and its analytical solutions. Computers & Industrial Engineering, 48(1):129-140, 2005.
-[53] Tang, L. C., & Xu, K. A unified approach for dual response surface optimization. Journal of quality technology, 34(4), 437-447, 2002.
+[53] Tang L.C., Xu K. A unified approach for dual response surface optimization. Journal of quality technology, 34(4):437-447, 2002.
-[54]  Steenackers, G., & Guillaume, P. Bias-specified robust design optimization: A generalized mean squared error approach. Computers & Industrial Engineering, 54(2), 259-268, 2008.
+[54]  Steenackers G.,  Guillaume P. Bias-specified robust design optimization: A generalized mean squared error approach. Computers & Industrial Engineering, 54(2):259-268, 2008.
-[55] Mandal W A. Weighted Tchebycheff optimization technique under uncertainty. Annals of Data Science, 1-23, 2020.
+[55] Mandal W.A. Weighted Tchebycheff optimization technique under uncertainty. Annals of Data Science, 8:709–731, 2021.
 [56] Dächert K, Gorski J, Klamroth K. An augmented weighted Tchebycheff method with adaptively chosen parameters for discrete bicriteria optimization problems. Computers & Operations Research, 39(12): 2929-2943, 2012.

@@ Line 1,355: / Line 1,355: @@
 [30] Shoham Y. Computer science and game theory. Communications of the ACM, 51(8):74-79, 2008.
-[31] Manshaei M H, Zhu Q, Alpcan T, et al. Game theory meets network security and privacy. ACM Computing Surveys (CSUR), 45(3): 1-39, 2013.
+[31] Manshaei M.H., Zhu Q., Alpcan T., et al. Game theory meets network security and privacy. ACM Computing Surveys (CSUR), 45(3):1-39, 2013.
-[32] Pillai P S, Rao S. Resource allocation in cloud computing using the uncertainty principle of game theory. IEEE Systems Journal, 10(2): 637-648, 2014.
+[32] Pillai P.S., Rao S. Resource allocation in cloud computing using the uncertainty principle of game theory. IEEE Systems Journal, 10(2):637-648, 2014.
-[33] Lemaire J. An application of game theory: cost allocation. ASTIN Bulletin: The Journal of the IAA. 1984; 14(1): 61-81, 2014.
+[33] Lemaire J. An application of game theory: cost allocation. ASTIN Bulletin: The Journal of the IAA, 1984; 14(1):61-81, 1984. Published online by Cambridge University Press in 2014.
-[34] Barough A S, Shoubi M V, Skardi M J E. Application of game theory approach in solving the construction project conflicts. Procedia-Social and Behavioral Sciences'','' 58: 1586-1593, 2012.
+[34] Barough A.S., Shoubi M.V., Skardi M.J.E. Application of game theory approach in solving the construction project conflicts. Procedia-Social and Behavioral Sciences, 58:1586-1593, 2012.
-[35] Gale D, Kuhn H W, Tucker A W. Linear programming and the theory of game. Activity analysis of production and allocation, 13: 317-335, 1951.
+[35] Gale D., Kuhn H.W., Tucker A.W. Linear programming and the theory of game. Activity analysis of production and allocation, 13: 317-335, 1951.
 [36] Mangasarian O L, Stone H. Two-person nonzero-sum games and quadratic programming. Journal of mathematical analysis and applications, 9(3): 348-355, 1964.

@@ Line 1,282: / Line 1,282: @@
 In general, for both cases, an increase in the value of <math display="inline">{d}_{i}</math> will increase the corresponding bargaining solution value. For example, an increasement of <math display="inline">{d}_{A}</math> will lead to an increase in process bias and a decrease in the variability value. This conclusion also makes sense from the perspective of game theory since it can be explained as disagreement point monotonicity [60] which can be defined as:<span id="cite-48"></span>
-For two points, <math display="inline">\, d=\left( {d}_{A},\, {\, d}_{B}\right)</math>  and <math display="inline">\, {d}^{'}=</math><math>\left( {{d}_{A}}^{'},\, {{\, d}_{B}}^{'}\right)</math> , <math display="inline">if\, {d}_{i}^{'}\geq {d}_{i},\, {d}_{j}^{,}={d}_{j},</math>then <math>{f}_{i}\left( U,{d}^{'}\right) \geq {f}_{i}\left( U,d\right) \, \, </math>; <math>j\not =i,\, and\, i,j\in \lbrace A,\, B\rbrace</math>
+For two points, <math display="inline">\, d=\left( {d}_{A},\, {d}_{B}\right)</math>  and <math display="inline"> {d}^{'}=\left( {d}_{A}^{'},\, d_{B}^{'}\right)</math>, <math display="inline">\mbox{if}\, {d}_{i}^{'}\geq {d}_{i},\, {d}_{j}^{,}={d}_{j},</math> then <math>{f}_{i}\left( U,{d}^{'}\right) \geq {f}_{i}\left( U,d\right)</math>; <math>j\not =i</math>, and <math> i,j\in \lbrace A,\, B\rbrace</math>, where <math display="inline">{f}_{i}\left( U,{d}^{'}\right)</math> and <math display="inline">{f}_{i}\left( U,d\right)</math>  represent the solution payoff for player ''i'' after and before the incensement of his disagreement point payoff, respectively. More specifically, the more disagreement point value (<math display="inline">{d}_{i}</math>) a player demands for participation in an agreement, the more the player will get. Although, a gain achieved by one player comes at the expense of the other player. This is because if the agreed solution is not an improvement for one player, then the player would not have any incentive to participate in the bargaining game. However, in the RPD case, the objective for a player is to minimize instead of maximize the utility value, so the less <math display="inline">{d}_{i}</math> a player proposes, the higher the requirement the player is actually proposing to participate in a bargaining game.
-−
 where <math display="inline">{f}_{i}\left( U,{d}^{'}\right)</math> and <math display="inline">{f}_{i}\left( U,d\right)</math>  represent the solution payoff for player ''i'' after and before the incensement of his disagreement point payoff, respectively. More specifically, the more disagreement point value ( <math display="inline">{d}_{i}</math>) a player demands for participation in an agreement, the more the player will get. Although, a gain achieved by one player comes at the expense of the other player. This is because if the agreed solution is not an improvement for one player, then the player would not have any incentive to participate in the bargaining game. However, in the RPD case, the objective for a player is to minimize instead of maximize the utility value, so the less <math display="inline">{d}_{i}</math> a player proposes, the higher the requirement the player is actually proposing to participate in a bargaining game.
 ==6. Conclusion and future direction ==

@@ Line 1,037: / Line 1,037: @@
 ===5.4 Sensitivity analysis for numerical example 2===
-Applying the same logic for example 2, <math display="inline">{d}_{B}</math> is kept constant as <math display="inline">{d}_{A}</math> is changed by 10%. [[#tab-7|Table 7]] exhibits the effect of changes in <math display="inline">{d}_{A}</math>, and Figure 9 demonstrates the efficiency of the calculated solutions.
+Applying the same logic for example 2, <math display="inline">{d}_{B}</math> is kept constant as <math display="inline">{d}_{A}</math> is changed by 10%. [[#tab-7|Table 7]] exhibits the effect of changes in <math display="inline">{d}_{A}</math>, and [[#img-9|Figure 9]] demonstrates the efficiency of the calculated solutions.
 <div class="center" style="font-size: 75%;">'''Table 7'''. Sensitivity analysis results for numerical example 2 by changing <math display="inline">{d}_{A}</math></div>
@@ Line 1,046: / Line 1,046: @@
 !<math>{d}_{A}</math> !! <math>{d}_{B}</math> !! <math>\left(\left( \hat{\mu }({\bf x}) -\tau \right)^2 -{d}_{A}\right)\ast \left( \hat{\sigma }^2({\bf x}) -{d}_{B}\right)</math> !! <math>x</math> !! <math>\left( \hat{\mu} ( {\bf x}^*) -\tau \right)^2</math> !! <math>\hat{\sigma}^2 ({\bf x}^*)</math>
 |-
-|  style="border-top: 1pt solid black;text-align: center;"|37.2266
+|  style="text-align: center;"|37.2266
-|  style="border-top: 1pt solid black;text-align: center;"|112.0959
+|  style="text-align: center;"|112.0959
-|  style="border-top: 1pt solid black;text-align: center;"|790.0487
+|  style="text-align: center;"|790.0487
-|  style="border-top: 1pt solid black;text-align: center;"|[ 0.9510    0.3554]
+|  style="text-align: center;"|[ 0.9510    0.3554]
-|  style="border-top: 1pt solid black;text-align: center;"|19.9778
+|  style="text-align: center;"|19.9778
-|  style="border-top: 1pt solid black;text-align: center;"|66.2928
+|  style="text-align: center;"|66.2928
 |-
 |  style="text-align: center;"|41.3629
@@ Line 1,137: / Line 1,137: @@
 |  style="text-align: center;"|53.7896
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;"|135.1396
+|  style="text-align: center;"|135.1396
-|  style="border-bottom: 1pt solid black;text-align: center;"|112.0959
+|  style="text-align: center;"|112.0959
-|  style="border-bottom: 1pt solid black;text-align: center;"|6204.7
+|  style="text-align: center;"|6204.7
-|  style="border-bottom: 1pt solid black;text-align: center;"|[1.0000    0.6179]
+|  style="text-align: center;"|[1.0000    0.6179]
-|  style="border-bottom: 1pt solid black;text-align: center;"|29.8115
+|  style="text-align: center;"|29.8115
-|  style="border-bottom: 1pt solid black;text-align: center;"|53.1879
+|  style="text-align: center;"|53.1879
 |}
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Draft_Shin_691882792-image9.png|centre|thumb|'''Figure 9.''' Plot of sensitivity analysis results with the Pareto frontier for numerical example 2 by changing
+<div id='img-9'></div>
-<math display="inline">{d}_{A}</math>
+{| style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: auto;max-width: auto;"
-|445x445px]]</div>
+|-
-On the other hand, another sensitivity analysis is conducted by changing <math display="inline">{d}_{B}</math> with 10% increment and decrement while holding <math display="inline">{d}_{A}</math> at a fixed value (63.0436) as shown in Table 8 and plotted in Figure 10.
+On the other hand, another sensitivity analysis is conducted by changing <math display="inline">{d}_{B}</math> with 10% increment and decrement while holding <math display="inline">{d}_{A}</math> at a fixed value (63.0436) as shown in [[#tab-8|Table 8]] and plotted in [[#img-10|Figure 10]].
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<div class="center" style="font-size: 75%;">'''Table 8'''. Sensitivity analysis results for numerical example 2 by changing <math display="inline">{d}_{B}</math></div>
-'''Table 8.''' Sensitivity analysis results for numerical example 2 by changing <math display="inline">{d}_{B}</math></div>
-{| style="width: 100%;border-collapse: collapse;"
+<div id='tab-8'></div>
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{d}_{A}</math>
+|  style="text-align: center;vertical-align: top;"|63.0436
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{d}_{B}</math>
+|  style="text-align: center;vertical-align: top;"|48.2536
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{\mathit{\boldsymbol{((}}\hat{\mu }\left( {\boldsymbol{\text{x}}}\right) \mathit{\boldsymbol{-}}\tau \mathit{\boldsymbol{)}}}^{\mathit{\boldsymbol{2}}}-</math><math>{d}_{A}\boldsymbol{)\ast (}{\hat{\sigma }}^{\mathit{\boldsymbol{2}}}\left( {\boldsymbol{\text{x}}}\right) -</math><math>{d}_{B}\mathit{\boldsymbol{)}}</math>
+|  style="text-align: center;vertical-align: top;"|15.4253
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{\boldsymbol{\text{x}}}</math>
+|  style="text-align: center;vertical-align: top;"|[1.0000    0.9166]
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;"|<math display="block">{\boldsymbol{(}\hat{\mu }\left( {\boldsymbol{\text{x}}}^*\right) \boldsymbol{-}\tau \boldsymbol{)}}^{{2}}</math>
+|  style="text-align: center;vertical-align: top;"|52.7429
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;text-align: center;"|<math>{\hat{\sigma }}^{\mathit{{2}}}\left( {\boldsymbol{\text{x}}}^*\right)</math>
+|  style="text-align: center;vertical-align: top;"|46.7561
-|-
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|63.0436
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|48.2536
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|15.4253
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|[1.0000    0.9166]
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|52.7429
-|  style="border-top: 1pt solid black;text-align: center;vertical-align: top;"|46.7561
 |-
 |  style="text-align: center;vertical-align: top;"|63.0436
@@ Line 1,264: / Line 1,262: @@
 |  style="text-align: center;vertical-align: top;"|66.2698
 |-
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|63.0436
+|  style="text-align: center;vertical-align: top;"|63.0436
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|198.5865
+|  style="text-align: center;vertical-align: top;"|198.5865
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|5708.3
+|  style="text-align: center;vertical-align: top;"|5708.3
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|[0.9199    0.3472]
+|  style="text-align: center;vertical-align: top;"|[0.9199    0.3472]
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|18.7938
+|  style="text-align: center;vertical-align: top;"|18.7938
-|  style="border-bottom: 1pt solid black;text-align: center;vertical-align: top;"|69.5842
+|  style="text-align: center;vertical-align: top;"|69.5842
 |}
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">[[File:Draft_Shin_691882792-image10.png|centre|423x423px|thumb|'''Figure 10.''' Plot of sensitivity analysis results with the Pareto frontier for numerical example 2 by changing <math display="inline">{d}_{B}</math>]]</div>
+<div id='img-10'></div>
-In general, for<span id="OLE_LINK13"></span><span id="OLE_LINK14"></span> both cases, an increase in the value of <math display="inline">{d}_{i}</math> will increase the corresponding bargaining solution value. For example, an increasement of <math display="inline">{d}_{A}</math> will lead to an increase in process bias and a decrease in the variability value. This conclusion also makes sense from the perspective of game theory since it can be explained as disagreement point monotonicity [60] which can be defined as:<span id="cite-48"></span>
+In general, for both cases, an increase in the value of <math display="inline">{d}_{i}</math> will increase the corresponding bargaining solution value. For example, an increasement of <math display="inline">{d}_{A}</math> will lead to an increase in process bias and a decrease in the variability value. This conclusion also makes sense from the perspective of game theory since it can be explained as disagreement point monotonicity [60] which can be defined as:<span id="cite-48"></span>
 For two points, <math display="inline">\, d=\left( {d}_{A},\, {\, d}_{B}\right)</math>  and <math display="inline">\, {d}^{'}=</math><math>\left( {{d}_{A}}^{'},\, {{\, d}_{B}}^{'}\right)</math> , <math display="inline">if\, {d}_{i}^{'}\geq {d}_{i},\, {d}_{j}^{,}={d}_{j},</math>then <math>{f}_{i}\left( U,{d}^{'}\right) \geq {f}_{i}\left( U,d\right) \, \, </math>; <math>j\not =i,\, and\, i,j\in \lbrace A,\, B\rbrace</math>

@@ Line 615: / Line 615: @@
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
-!<math>{d}_{A}</math> !! <math>{d}_{B}</math> !! <math>\left(\left( \hat{\mu }({\bf x}) -t \right)^2 -{d}_{A}\right)\ast \left( \hat{\sigma }({\bf x}) -{d}_{B}\right)</math> !! <math>x</math> !! <math>\left( \mu ({\bf x}^*) -\tau \right)^2</math> !! <math>\sigma ({\bf x}^*)</math>
+!<math>{d}_{A}</math> !! <math>{d}_{B}</math> !! <math>\left(\left( \hat{\mu }({\bf x}) -\tau \right)^2 -{d}_{A}\right)\ast \left( \hat{\sigma }({\bf x}) -{d}_{B}\right)</math> !! <math>x</math> !! <math>\left( \mu ({\bf x}^*) -\tau \right)^2</math> !! <math>\sigma ({\bf x}^*)</math>
 |-
 |  style="text-align: center;"|0.6589
@@ Line 735: / Line 735: @@
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
-!<math>{d}_{A}</math> !! <math>{d}_{B}</math> !! <math>\left(\left( \hat{\mu }({\bf x}) -t \right)^2 -{d}_{A}\right)\ast \left( \hat{\sigma }({\bf x}) -{d}_{B}\right)</math> !! <math>x</math> !! <math>\left( \hat{\mu} ( {\bf x}^*) -\tau \right)^2</math> !! <math>\hat{\sigma} ({\bf x}^*)</math>
+!<math>{d}_{A}</math> !! <math>{d}_{B}</math> !! <math>\left(\left( \hat{\mu }({\bf x}) -\tau \right)^2 -{d}_{A}\right)\ast \left( \hat{\sigma }({\bf x}) -{d}_{B}\right)</math> !! <math>x</math> !! <math>\left( \hat{\mu} ( {\bf x}^*) -\tau \right)^2</math> !! <math>\hat{\sigma} ({\bf x}^*)</math>
 |-
 |  style="text-align: center;"|1.2398

@@ Line 574: / Line 574: @@
 {| class="wikitable" style="margin: 1em auto 0.1em auto;border-collapse: collapse;font-size:85%;width:auto;"
 |-style="text-align:center"
-! !! <math>{x}_{1}^{\ast }</math> !! <math>{x}_{2}^{\ast }</math> !! <math>{x}_{3}^{\ast }</math> !! <math>{\left( \hat{\mu }\left( {\boldsymbol{\text{x}}}^*\right) -\tau \right) }^{2}</math> !! <math>{\hat{\sigma }}^{2}\mathit{\boldsymbol{(}}{\boldsymbol{\text{x}}}^*\mathit{\boldsymbol{)}}</math> !! MSE
+! !! <math>{x}_{1}^{\ast }</math> !! <math>{x}_{2}^{\ast }</math> !! <math>{x}_{3}^{\ast }</math> !! <math>{\left( \hat{\mu }\left( {\bf x}^*\right) -\tau \right) }^{2}</math> !! <math>\hat{\sigma }^{2}({\bf x}^*)</math> !! MSE
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 |  style="text-align: left;"|'''Dual response model with WLS'''