The Quasi-Normal Direction (QND) Method: An Efficient Method for Finding the Pareto Frontier in Multi-Objective Optimization Problems

Document Type : Research Paper

Author

Department of Mathematical sciences, Islamic Azad University, Lahijan Branch

Abstract

In managerial and economic applications, there appear problems in which the goal is to simultaneously optimize several criteria functions (CFs). However, since the CFs are in conflict with each other in such cases, there is not a feasible point available at which all CFs could be optimized simultaneously. Thus, in such cases, a set of points, referred to as 'non-dominate' points (NDPs), will be encountered that are ineffective in relation to each other. In order to find such NDPs, many methods including the scalarization techniques have been proposed, each with their advantages and disadvantages. A comprehensive approach with scalarization perspective is the PS method of Pascoletti and Serafini. The PS method uses the two parameters of  as the starting point and  as the direction of motion to find the NDPs on the 'non-dominate' frontier (NDF). In bi-objective cases, the point  is selected on a special line, and changing point on this line leads to finding all the NDPs. Generalization of this approach is very difficult to three- or more-criteria optimization problems because any closed pointed cone in a three- or more-dimensional space is not like a two-dimensional space of a polygonal cone. Moreover, even for multifaceted cones, the method cannot be generalized, and inevitably weaker constraints must be used in the assumptions of the method. In order to overcome such problems of the PS method, instead of a hyperplane (two-dimensional line), a hypersphere is applied in the current paper, and the parameter  is changed over its boundary. The generalization of the new method for more than two criteria problems is simply carried out, and the examples, provided along with their comparisons with methods such as mNBI and NC, ensure the efficiency of the method. A case study in the realm of health care management (HCM) including two conflicting CFs with special constraints is also presented as an exemplar application of the proposed method.

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References

Abo-Sinna, M., Abo-Elnaga, Y. Y., and Mousa, A. (2014). An interactive dynamic approach based on hybrid swarm optimization for solving multiobjective programming problem with fuzzy parameters. Applied Mathematical Modelling, 38(7-8), 2000-2014.
Alber, M., and Reemtsen, R. (2007). Intensity modulated radiotherapy treatment planning by use of a barrier-penalty multiplier method. Optimisation Methods and Software, 22(3), 391-411.
Audet, C., Savard, G., and Zghal, W. (2008). Multiobjective optimization through a series of single-objective formulations. SIAM Journal on Optimization, 19(1), 188-210.
Brahme, A. (1984). Dosimetric precision requirements in radiation therapy. Acta Radiologica: Oncology, 23(5), 379-391.
Cotrutz, C., Lahanas, M., Kappas, C., and Baltas, D. (2001). A multiobjective gradient-based dose optimization algorithm for external beam conformal radiotherapy. Physics in Medicine and Biology, 46(8), 2161.
Craft, D., Halabi, T., Shih, H. A., and Bortfeld, T. (2007). An approach for practical multiobjective IMRT treatment planning. International journal of radiation oncology* Biology* Physics, 69(5), 1600-1607.
Deb, K. (2001). Multi-objective optimization using evolutionary algorithms (Vol. 16). John Wiley and Sons.
Ehrgott, M., and Burjony, M. (2001). Radiation therapy planning by multicriteria optimization. In Proceedings of the 36th Annual Conference of the Operational Research Society of New Zealand (pp. 244-253).
Eichfelder, G. (2008). Adaptive scalarization methods in multiobjective optimization (Vol. 436). Berlin: Springer.
Eichfelder, G. (2009). Scalarizations for adaptively solving multi-objective optimization problems. Computational Optimization and Applications, 44(2), 249.
Eichfelder, G. (2014). Vector optimization in medical engineering Mathematics Without Boundaries (pp. 181-215). Springer, New York, NY.
Kasimbeyli, R., Ozturk, Z. K., Kasimbeyli, N., Yalcin, G. D., and Erdem, B. I. (2017). Comparison of Some Scalarization Methods in Multiobjective Optimization. Bulletin of the Malaysian Mathematical Sciences Society,  1-31.
Küfer, K.-H., Scherrer, A., Monz, M., Alonso, F., Trinkaus, H., Bortfeld, T., and Thieke, C. (2003). Intensity-modulated radiotherapy–a large scale multi-criteria programming problem. OR spectrum, 25(2), 223-249.
Lopeza, R. H., Rittob, T., Sampaioc, R., and de Cursid, J. E. S. (2013). A new multiobjective optimization algorithm for nonconvex pareto fronts and objective functions. Asociación Argentina de Mecánica Computacional, 669-679.
Meng, H.-y., Zhang, X.-h., and Liu, S.-y. (2005). In International Conference on Natural Computation (pp. 1044-1048). Springer, Berlin, Heidelberg.
Messac, A., Ismail-Yahaya, A., and Mattson, C. A. (2003). The normalized normal constraint method for generating the Pareto frontier. Structural and Multidisciplinary Optimization, 25(2), 86-98.
Messac, A., and Mattson, C. A. (2004). Normal constraint method with guarantee of even representation of complete Pareto frontier. AIAA journal, 42(10), 2101-2111.
Niemierko, A. (1997). Reporting and analyzing dose distributions: a concept of equivalent uniform dose. Medical physics, 24(1), 103-110.
Pardalos, P. M., Žilinskas, A., and Žilinskas, J. (2017). Non-convex multi-objective optimization. Springer International Publishing.
Pintér, J. D., Linder, D., and Chin, P. (2006). Global Optimization Toolbox for Maple: An introduction with illustrative applications. Optimisation Methods and Software, 21(4), 565-582.
Shukla, P. K. (2007). On the normal boundary intersection method for generation of efficient front. In International Conference on Computational Science (pp. 310-317). Springer, Berlin, Heidelberg.
Siddiqui, S., Azarm, S., and Gabriel, S. (2011). A modified Benders decomposition method for efficient robust optimization under interval uncertainty. Structural and Multidisciplinary Optimization, 44(2), 259-275.
Uilhoorn, F. E. (2017). Comparison of Bayesian estimation methods for modeling flow transients in gas pipelines. Journal of Natural Gas Science and Engineering, 38:159– 170.
Valipour, E., Yaghoobi, M., and Mashinchi, M. (2014). An iterative approach to solve multiobjective linear fractional programming problems. Applied Mathematical Modelling, 38(1), 38-49.
Zhang, Q., Zhou, A., Zhao, S., Suganthan, P. N., Liu, W., and Tiwari, S. (2008). Multiobjective optimization test instances for the CEC 2009 special session and competition. University of Essex, Colchester, UK and Nanyang technological University, Singapore, special session on performance assessment of multi-objective optimization algorithms.
Interdisciplinary Journal of Management Studies (Formerly known as Iranian Journal of Management Studies)
Volume 12, Issue 3
September 2019
Pages 379-404

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