# Solving multiobjective optimization problems with decision uncertainty: an interactive approach

- 171 Downloads

## Abstract

We propose an interactive approach to support a decision maker to find a most preferred robust solution to multiobjective optimization problems with decision uncertainty. A new robustness measure that is understandable for the decision maker is incorporated as an additional objective in the problem formulation. The proposed interactive approach utilizes elements of the synchronous NIMBUS method and is aimed at supporting the decision maker to consider the objective function values and the robustness of a solution simultaneously. In the interactive approach, we offer different alternatives for the decision maker to express her/his preferences related to the robustness of a solution. To consolidate the interactive approach, we tailor a visualization to illustrate both the objective function values and the robustness of a solution. We demonstrate the advantages of the interactive approach by solving example problems.

## Keywords

Multiple criteria decision making Robust solutions Interactive methods Handling uncertainties NIMBUS Robustness measure## JEL Classification

C61## Notes

### Acknowledgements

We thank Dr. Dmitry Podkopaev for various discussions in constructing the multiobjective version of the procurement contract selection and pricing optimization problem.

## References

- Asafuddoula M, Singh H, Ray T (2015) Six-sigma robust design optimization using a many-objective decomposition-based evolutionary algorithm. IEEE Trans Evol Comput 19(4):490–507Google Scholar
- Azaron A, Brown K, Tarim S, Modarres M (2008) A multi-objective stochastic programming approach for supply chain design considering risk. Int J Prod Econ 116(1):129–138Google Scholar
- Barrico C, Antunes C (2006) Robustness analysis in multi-objective optimization using a degree of robustness concept. Proc IEEE Congr Evol Comput CEC 2006:1887–1892Google Scholar
- Bertsimas D, Sim M (2004) The price of robustness. Oper Res 52(1):35–53Google Scholar
- Bokrantz R, Fredriksson A (2017) Necessary and sufficient conditions for Pareto efficiency in robust multiobjective optimization. Eur J Oper Res 262(2):682–692Google Scholar
- Branke J, Deb K, Miettinen K, Słowiński R (eds) (2008) Multiobjective optimization: interactive and evolutionary approaches. Springer, BerlinGoogle Scholar
- Calfa B, Grossmann I (2015) Optimal procurement contract selection with price optimization under uncertainty for process networks. Comput Chem Eng 82:330–343Google Scholar
- Deb K, Gupta H (2006) Introducing robustness in multi-objective optimization. Evol Comput 14(4):463–494Google Scholar
- Ehrgott M, Ide J, Schöbel A (2014) Minmax robustness for multi-objective optimization problems. Eur J Oper Res 239(1):17–31Google Scholar
- Eichfelder G, Krüger C, Schöbel A (2017) Decision uncertainty in multiobjective optimization. J Glob Optim 69(2):485–510Google Scholar
- Gaspar-Cunha A, Covas JA (2007) Robustness in multi-objective optimization using evolutionary algorithms. Comput Optim Appl 39(1):75–96Google Scholar
- Gunawan S, Azarm S (2005) Multi-objective robust optimization using a sensitivity region concept. Struct Multidiscip Optim 29(1):50–60Google Scholar
- Hassanzadeh E, Nemati H, Sun M (2013). Robust optimization for multiobjective programming problems with imprecise information. Proc Comput Sci 17:357–364. First International Conference on Information Technology and Quantitative ManagementGoogle Scholar
- Ide J, Schöbel A (2016) Robustness for uncertain multi-objective optimization: a survey and analysis of different concepts. OR Spectr 38(1):235–271Google Scholar
- Li M, Azarm S, Aute V (2005) A multi-objective genetic algorithm for robust design optimization. In : Proceedings of the 7th annual conference on genetic and evolutionary computation, GECCO 2005, pp 771–778Google Scholar
- Liang Y, Cheng X, Li Z, Xiang J (2011) Robust multi-objective wing design optimization via CFD approximation model. Eng Appl Comput Fluid Mech 5(2):286–300Google Scholar
- Miettinen K (1999) Nonlinear multiobjective optimization. Kluwer Academic Publishers, BostonGoogle Scholar
- Miettinen K (2006) IND-NIMBUS for demanding interactive multiobjective optimization. In: Trzaskalik T (ed) Multiple criteria decision making ’05. The Karol Adamiecki University of Economics in Katowice, Katowice, pp 137–150Google Scholar
- Miettinen K, Mäkelä MM (2002) On scalarizing functions in multiobjective optimization. OR Spectr 24(2):193–213Google Scholar
- Miettinen K, Mäkelä MM (2006) Synchronous approach in interactive multiobjective optimization. Eur J Oper Res 170(3):909–922Google Scholar
- Miettinen K, Mustajoki J, Stewart TJ (2014) Interactive multiobjective optimization with NIMBUS for decision making under uncertainty. OR Spectrum 36(1):39–56Google Scholar
- Narula S, Weistroffer H (1989) A flexible method for nonlinear multicriteria decision-making problems. IEEE Trans Syst Man Cybern 19(4):883–887Google Scholar
- Ojalehto V, Miettinen K, Laukkanen T (2014) Implementation aspects of interactive multiobjective optimization for modeling environments: the case of GAMS-NIMBUS. Comput Optim Appl 58(3):757–779Google Scholar
- Phillips R (2005) Pricing and revenue optimization. Stanford University Press, Palo AltoGoogle Scholar
- Salimi A, Lowther DA (2016) On the role of robustness in multi-objective robust optimization: application to an IPM motor design problem. IEEE Trans Magn 52(3):1–4Google Scholar
- Sawaragi Y, Nakayama H, Tanino T (1985) Theory of multiobjective optimization. Academic Press Inc., CambridgeGoogle Scholar
- Steuer R (1986) Multiple criteria optimization: theory, computation and applications. Wiley, New YorkGoogle Scholar
- Sun G, Li G, Zhou S, Li H, Hou S, Li Q (2010) Crashworthiness design of vehicle by using multiobjective robust optimization. Struct Multidiscip Optim 44(1):99–110Google Scholar
- Talaei M, Moghaddam BF, Pishvaee MS, Bozorgi-Amiri A, Gholamnejad S (2016) A robust fuzzy optimization model for carbon-efficient closed-loop supply chain network design problem: a numerical illustration in electronics industry. J Clean Prod 113:662–673Google Scholar
- Wiecek MM, Dranichak GM (2016) Robust multiobjective optimization for decision making under uncertainty and conflict. In: Gupta A, Capponi A, Smith JC, Greenberg HJ (eds) Optimization challenges in complex, networked and risky systems. Springer, New York, pp 84–114Google Scholar
- Wiecek MM, Blouin VY, Fadel GM, Engau A, Hunt BJ, Singh V (2009) Multi-scenario multi-objective optimization with applications in engineering design. In: Barichard V, Ehrgott M, Gandibleux X, T’Kindt V (eds) Multiobjective programming and goal programming: theoretical results and practical applications. Springer, Berlin, pp 283–298Google Scholar
- Wierzbicki AP (1982) A mathematical basis for satisficing decision making. Math Model 3(5):391–405Google Scholar
- Yeh W-C, Chuang M-C (2011) Using multi-objective genetic algorithm for partner selection in green supply chain problems. Expert Syst Appl 38(4):4244–4253Google Scholar