Easy-Interactive Ordering of the Pareto Optimal Set with Imprecise Weights
In the multi objective optimization, in the case when generated set of Pareto optimal solutions is large, occurs the problem to select of the best solution from this set. In this paper, is suggested a method to order of Pareto set. Ordering the Pareto optimal set carried out in conformity with the introduced distance function between each solution and selected reference point, where the reference point may be adjusted to represent the preferences of a decision making agent. Preference information about objective weights from a decision maker may be expressed imprecisely. The developed elicitation procedure provides an opportunity to obtain surrogate numerical weights for the objectives, and thus, to manage impreciseness of preference. The proposed method is a scalable to many objectives and can be used independently or as complementary to the various visualization techniques in the multidimensional case.
[1] I. Das, "A preference ordering among various Pareto optimal
alternatives," Structural and Multidisciplinary Optimization, vol. 18, no.
1, pp. 30-35, 1999a.
[2] G. K. Kao and S. H. Jacobson, "Finding preferred subsets of Pareto
optimal solutions," Computational Optimization and Applications, vol.
40, no. 1, pp. 73-95, 2008.
[3] C. A. Mattson, A. A. Mullur and A. Messac, "Smart Pareto filter:
obtaining a minimal representation of multiobjective design space,"
Engineering Optimization, vol. 36, no. 6, pp. 721-740, 2004.
[4] M. Farrow and M. Goldstein, "Sensitivity of decisions with imprecise
utility trade-off parameters using boundary linear utility," International
Journal of Approximate Reasoning, vol. 51, no. 9, pp. 1100-1113, 2010.
[5] V. Venkat, S. H. Jacobson and J. A. Stori , "A Post-Optimality Analysis
Algorithm for Multi-Objective Optimization," Computational
Optimization and Applications, vol. 28, no. 3, pp. 357-372 , 2004.
[6] T. Aittokoski, S. ├äyr├ñmö and K. Miettinen , "Clustering aided approach
for decision making in computationally expensive multiobjective
optimization," Optimization Methods and Software, vol. 24, no. 2, pp.
157-174, 2009.
[7] A. V. Lotov and K. Miettinen, "Visualizing the Pareto Frontier," in
Multiobjective Optimization, Berlin Heidelberg, Springer-Verlag, 2008,
pp. 213-243.
[8] P. Korhonen and J. Wallenius, "Visualization in the Multiple Objective
Decision-Making Framework," in Multiobjective Optimization, Berlin /
Heidelberg, Springer, 2008, pp. 195-212.
[9] R. Efremov, D. R. Insua and A. Lotov, "A framework for participatory
decision support using Pareto frontier visualization, goal identification
and arbitration," European Journal of Operational Research, vol. 199,
no. 2, pp. 459-467, 2009.
[10] X. Bi and B. Li, "The visualization decision-making model of four
objectives based on the balance of space vector," Nanchang, Jiangxi,
2012.
[11] K. Deb, Multi-Objective Optimization using Evolutionary Algorithms,
United Kingdom: John Wiley & Sons, Ltd, 2009.
[12] K. Miettinen, "Introduction to Multiobjective Optimization:
Noninteractive Approach," in Multiobjective Optimization, Berlin
Heidelberg, Springer-Verlag, 2008, pp. 1-26.
[13] A. V. Lotov and K. Miettinen, "Visualizing the Pareto Frontier," in
Multiobjective Optimization, Berlin Heidelberg, Springer-Verlag, 2008,
pp. 213-243.
[14] K. Miettinen, "Introduction to Multiobjective Optimization:
Noninteractive Approach," i Multiobjective Optimization, Berlin
Heidelberg, Springer-Verlag, 2008, pp. 1-26.
[15] I. Das, "On characterizing the "knee" of the Pareto curve based on
Normal-Boundary Intersection," Structural and Multidisciplinary
Optimization, vol. 18, no. 2, pp. 107-115, 1999b.
[16] M. Riabacke, M. Danielson och L. Ekenberg, "State-of-the-art in
prescriptive criteria weight elicitation," Advances in Decision Sciences,
2012.
[17] T. L. Saaty, The Analytic Hierarcy Process, McGraw Hill International,
1980.
[18] H. F. Barron, "Selecting a Best Multiattribute Alternative with Partial
Information about Attribut Weights," Acta Psychologica, vol. 80, no. 1-
3, pp. 91-103, 1992.
[19] H. F. Barron och B. E. Barrett, "Decision quality using ranked attribute
weights," Management Science, vol. 42, nr 11, pp. 1515-1523, 1996.
[20] M. Kalinina, A. Larsson and L. Olsson, "Generating and Ordering of
Transport Alternatives in Inter-Modal Logistics in the Presence of Cost,
Time, and Emission Conflicts," 2012 IEEE Int. Conference on Industrial
Engineering and Engineering Management, 2012.
[1] I. Das, "A preference ordering among various Pareto optimal
alternatives," Structural and Multidisciplinary Optimization, vol. 18, no.
1, pp. 30-35, 1999a.
[2] G. K. Kao and S. H. Jacobson, "Finding preferred subsets of Pareto
optimal solutions," Computational Optimization and Applications, vol.
40, no. 1, pp. 73-95, 2008.
[3] C. A. Mattson, A. A. Mullur and A. Messac, "Smart Pareto filter:
obtaining a minimal representation of multiobjective design space,"
Engineering Optimization, vol. 36, no. 6, pp. 721-740, 2004.
[4] M. Farrow and M. Goldstein, "Sensitivity of decisions with imprecise
utility trade-off parameters using boundary linear utility," International
Journal of Approximate Reasoning, vol. 51, no. 9, pp. 1100-1113, 2010.
[5] V. Venkat, S. H. Jacobson and J. A. Stori , "A Post-Optimality Analysis
Algorithm for Multi-Objective Optimization," Computational
Optimization and Applications, vol. 28, no. 3, pp. 357-372 , 2004.
[6] T. Aittokoski, S. ├äyr├ñmö and K. Miettinen , "Clustering aided approach
for decision making in computationally expensive multiobjective
optimization," Optimization Methods and Software, vol. 24, no. 2, pp.
157-174, 2009.
[7] A. V. Lotov and K. Miettinen, "Visualizing the Pareto Frontier," in
Multiobjective Optimization, Berlin Heidelberg, Springer-Verlag, 2008,
pp. 213-243.
[8] P. Korhonen and J. Wallenius, "Visualization in the Multiple Objective
Decision-Making Framework," in Multiobjective Optimization, Berlin /
Heidelberg, Springer, 2008, pp. 195-212.
[9] R. Efremov, D. R. Insua and A. Lotov, "A framework for participatory
decision support using Pareto frontier visualization, goal identification
and arbitration," European Journal of Operational Research, vol. 199,
no. 2, pp. 459-467, 2009.
[10] X. Bi and B. Li, "The visualization decision-making model of four
objectives based on the balance of space vector," Nanchang, Jiangxi,
2012.
[11] K. Deb, Multi-Objective Optimization using Evolutionary Algorithms,
United Kingdom: John Wiley & Sons, Ltd, 2009.
[12] K. Miettinen, "Introduction to Multiobjective Optimization:
Noninteractive Approach," in Multiobjective Optimization, Berlin
Heidelberg, Springer-Verlag, 2008, pp. 1-26.
[13] A. V. Lotov and K. Miettinen, "Visualizing the Pareto Frontier," in
Multiobjective Optimization, Berlin Heidelberg, Springer-Verlag, 2008,
pp. 213-243.
[14] K. Miettinen, "Introduction to Multiobjective Optimization:
Noninteractive Approach," i Multiobjective Optimization, Berlin
Heidelberg, Springer-Verlag, 2008, pp. 1-26.
[15] I. Das, "On characterizing the "knee" of the Pareto curve based on
Normal-Boundary Intersection," Structural and Multidisciplinary
Optimization, vol. 18, no. 2, pp. 107-115, 1999b.
[16] M. Riabacke, M. Danielson och L. Ekenberg, "State-of-the-art in
prescriptive criteria weight elicitation," Advances in Decision Sciences,
2012.
[17] T. L. Saaty, The Analytic Hierarcy Process, McGraw Hill International,
1980.
[18] H. F. Barron, "Selecting a Best Multiattribute Alternative with Partial
Information about Attribut Weights," Acta Psychologica, vol. 80, no. 1-
3, pp. 91-103, 1992.
[19] H. F. Barron och B. E. Barrett, "Decision quality using ranked attribute
weights," Management Science, vol. 42, nr 11, pp. 1515-1523, 1996.
[20] M. Kalinina, A. Larsson and L. Olsson, "Generating and Ordering of
Transport Alternatives in Inter-Modal Logistics in the Presence of Cost,
Time, and Emission Conflicts," 2012 IEEE Int. Conference on Industrial
Engineering and Engineering Management, 2012.
@article{"International Journal of Information, Control and Computer Sciences:52476", author = "Maria Kalinina and Aron Larsson and Leif Olsson", title = "Easy-Interactive Ordering of the Pareto Optimal Set with Imprecise Weights", abstract = "In the multi objective optimization, in the case when generated set of Pareto optimal solutions is large, occurs the problem to select of the best solution from this set. In this paper, is suggested a method to order of Pareto set. Ordering the Pareto optimal set carried out in conformity with the introduced distance function between each solution and selected reference point, where the reference point may be adjusted to represent the preferences of a decision making agent. Preference information about objective weights from a decision maker may be expressed imprecisely. The developed elicitation procedure provides an opportunity to obtain surrogate numerical weights for the objectives, and thus, to manage impreciseness of preference. The proposed method is a scalable to many objectives and can be used independently or as complementary to the various visualization techniques in the multidimensional case.
", keywords = "Imprecise weights, Multiple objectives, Pareto
optimality, Visualization.", volume = "7", number = "4", pages = "476-6", }
