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An approach to identify solutions of interest from multi and many-objective optimization problems

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Abstract

The result of a multiobjective or a many-objective optimization problem is a large set of non-dominated solutions. Once the Pareto Front (or a good approximation of it) has been found, then providing the decision maker with a smaller set of “interesting solutions” is a key step. Here, the focus is on how to select such a set of solutions of interest which, in contrast to previous approaches that relied on geometrical features, is carried out considering the decision maker’s preferences. The proposed a posteriori approach consists in assigning an interval of potential scores to every solution, where such scores depend on the decision maker’s preferences. The solutions are then compared and filtered according to their corresponding intervals, using a recently proposed possibility degree formula. Three examples, with two, three and many objectives are used to show the benefits of the proposal.

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Notes

  1. For certain attitude functions like \(f(x)=1/x\) the intervals must be defined in \(R^+\).

  2. The benchmark provided in the Competition on Many-Objective Optimization at the 2018 IEEE Congress on Evolutionary Computation, included functions with 5, 10 and 15 objectives https://www.cs.bham.ac.uk/~chengr/CEC_Comp_on_MaOO/2018/webpage.html.

  3. In the special case where this interval is not unique, the reference interval is the one that also has the greatest upper bound.

References

  1. Ahn BS, Park KS (2008) Comparing methods for multiattribute decision making with ordinal weights. Comput Oper Res 35(5):1660–1670 Part special issue: algorithms and computational methods in feasibility and infeasibility

    Article  Google Scholar 

  2. Barron FH, Barrett BE (1996) Decision quality using ranked attribute weights. Manag Sci 42(11):1515–1523

    Article  Google Scholar 

  3. Beliakov G, Pradera A, Calvo T (2007) Aggregation functions: a guide for practitioners, vol 221. Springer, Berlin

    MATH  Google Scholar 

  4. Bhattacharjee KS, Singh HK, Ray T (2016) A study on performance metrics to identify solutions of interest from a trade-off set. In: Australasian conference on artificial life and computational intelligence, pp. 66–77. Springer

  5. Bhattacharjee KS, Singh HK, Ryan M, Ray T (2017) Bridging the gap: many-objective optimization and informed decision-making. IEEE Trans Evol Comput 21(5):813–820

    Article  Google Scholar 

  6. Branke J (2016) MCDA and multiobjective evolutionary algorithms. In: Greco S, Ehrgott M, Figueira JR (eds) Multiple criteria decision analysis: state of the art surveys. Springer, New York, pp 977–1008

    Chapter  Google Scholar 

  7. Branke J, Deb K, Dierolf H, Osswald M et al (2004) Finding knees in multi-objective optimization. PPSN 3242:722–731

    Google Scholar 

  8. Coello CAC, Lamont GB, Van Veldhuizen DA (2007) Evolutionary algorithms for solving multi-objective problems, vol 5. Genetic and evolutionary computation series. Springer, Boston

    MATH  Google Scholar 

  9. Cruz-Reyes L, Fernandez E, Sanchez P, Coello CAC, Gomez C (2017) Incorporation of implicit decision-maker preferences in multi-objective evolutionary optimization using a multi-criteria classification method. Appl Soft Comput 50:48–57

    Article  Google Scholar 

  10. Da Q-L, Liu X-W (1999) Interval number linear programming and its satisfactory solution. Syst Eng Theory Pract 19:3–7

    Google Scholar 

  11. Das I (1999) On characterizing the “knee” of the pareto curve based on normal-boundary intersection. Struct Multidiscip Optim 18(2):107–115

    Article  Google Scholar 

  12. Facchinetti G, Ricci RG, Muzzioli S (1998) Note on ranking fuzzy triangular numbers. Int J Intell Syst 13(7):613–622

    Article  Google Scholar 

  13. Hughes E (2019) Many-objective radar design software. http://code.evanhughes.org. Accessed 21 Mar 2019

  14. Karmakar S, Bhunia AK (2012) A comparative study of different order relations of intervals. Reliab Comput 16:38–72

    MathSciNet  MATH  Google Scholar 

  15. Larichev O (1992) Cognitive validity in design of decision-aiding techniques. J Multicriteria Decis Anal 1(3):127–138

    Article  Google Scholar 

  16. Levin V (2004) Ordering of intervals and optimization problems with interval parameters. Cybern Syst Anal 40(3):316–324

    Article  MathSciNet  Google Scholar 

  17. Li D, Zeng W, Yin Q (2017) Novel ranking method of interval numbers based on the Boolean matrix. Soft Comput 22:4113–4122

    Article  Google Scholar 

  18. Li K, Chen R, Min G, Yao X (2018) Integration of preferences in decomposition multiobjective optimization. IEEE Trans Cybern 48(12):3359–3370

    Article  Google Scholar 

  19. Liu F, Pan L-H, Liu Z-L, Peng Y-N (2018) On possibility-degree formulae for ranking interval numbers. Soft Comput 22(8):2557–2565

    Article  Google Scholar 

  20. López-Jaimes A, Coello CAC (2014) Including preferences into a multiobjective evolutionary algorithm to deal with many-objective engineering optimization problems. Inf Sci 277:1–20

    Article  MathSciNet  Google Scholar 

  21. Malhotra NK (1982) Information load and consumer decision making. J Consum Res 8(4):419–430

    Article  Google Scholar 

  22. Moore RE (1979) Methods and applications of interval analysis, vol 2. SIAM, Philadelphia

    Book  Google Scholar 

  23. Sengupta A, Pal TK (2000) On comparing interval numbers. Eur J Oper Res 127(1):28–43

    Article  MathSciNet  Google Scholar 

  24. Stillwell WG, Seaver DA, Edwards W (1981) A comparison of weight approximation techniques in multiattribute utility decision making. Organ Behav Hum Perform 28(1):62–77

    Article  Google Scholar 

  25. Sun H-L, Yao W-X (2010) Comments on methods for ranking interval numbers. J Syst Eng 3:005

    Google Scholar 

  26. Wang H, Olhofer M, Jin Y (2017) A mini-review on preference modeling and articulation in multi-objective optimization: current status and challenges. Complex Intell Syst 3(4):233–245

    Article  Google Scholar 

  27. Wang Y-M, Yang J-B, Xu D-L (2005) A two-stage logarithmic goal programming method for generating weights from interval comparison matrices. Fuzzy Sets Syst 152(3):475–498

    Article  MathSciNet  Google Scholar 

  28. Xu Y, Da Q (2008) A method for multiple attribute decision making with incomplete weight information under uncertain linguistic environment. Knowl Based Syst 21(8):837–841

    Article  Google Scholar 

  29. Xu Z (2007) A method for multiple attribute decision making with incomplete weight information in linguistic setting. Knowl Based Syst 20(8):719–725

    Article  Google Scholar 

  30. Xu Z, Da QL (2002) The uncertain OWA operator. Int J Intell Syst 17(6):569–575

    Article  Google Scholar 

  31. Xu Z, Da QL (2003) Possibility degree method for ranking interval numbers and its application. J Syst Eng 18:67–70

    Google Scholar 

  32. Zhang X, Tian Y, Jin Y (2015) A knee point-driven evolutionary algorithm for many-objective optimization. IEEE Trans Evol Comput 19(6):761–776

    Article  Google Scholar 

  33. Zhou L-G, Chen H-Y, Merigó JM, Gil-Lafuente AM (2012) Uncertain generalized aggregation operators. Expert Syst Appl 39(1):1105–1117

    Article  Google Scholar 

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Acknowledgements

D. A. Pelta and M. T. Lamata acknowledge support through Project TIN2017-86647-P from the Spanish Ministry of Economy and Competitiveness (including European Regional Development Funds). M. Torres enjoys a Ph.D. research training staff grant associated with the Project TIN2014-55024-P from the Spanish Ministry of Economy and Competitiveness and co-funded by the European Social Fund. R. Yager acknowledges the support of the United States Office of Naval Research (ONR).

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Authors and Affiliations

  1. Models of Decision and Optimization Research Group, Department of Computer Science and A.I., Universidad de Granada, 18014, Granada, Spain

    Marina Torres, David A. Pelta & María T. Lamata

  2. Machine Intelligence Institute Iona College, New Rochelle, NY, 10801, USA

    Ronald R. Yager

Authors
  1. Marina Torres
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  2. David A. Pelta
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  3. María T. Lamata
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  4. Ronald R. Yager
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Contributions

All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by MT and DAP. All authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.

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Correspondence to David A. Pelta.

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Torres, M., Pelta, D.A., Lamata, M.T. et al. An approach to identify solutions of interest from multi and many-objective optimization problems. Neural Comput & Applic 33, 2471–2481 (2021). https://doi.org/10.1007/s00521-020-05140-x

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  • DOI: https://doi.org/10.1007/s00521-020-05140-x

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