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mehrdad Ghaznavi Mohammad Ilati Esmaile Khorram

Abstract

‎The wide variety of available interactive methods brings the need for creating general‎ ‎interactive algorithms enabling the decision maker (DM) to apply freely several convenient methods which best fit his/her preferences‎. ‎To this end‎, ‎in this paper‎, ‎we propose a general scalarizing problem for multiobjective programming problems‎.
‎The relation between optimal solutions of the introduced scalarizing problem and (weakly) efficient as well as properly efficient solutions of the main multiobjective optimization problem (MOP) is discussed‎. ‎It is shown that some of the scalarizing problems used in different interactive methods can be obtained from proposed formulation by selecting suitable transformations‎. ‎Based on the suggested scalarizing problem‎, ‎we propose a general interactive algorithm (GIA) that enables the DM to specify his/her preferences in six different ways with capability to change his/her preferences any time during the iterations of the algorithm‎.
‎Finally‎, ‎a numerical example demonstrating the applicability of the algorithm is provided‎.

Article Details

References
1. Benayoun, R., Montgolfier, J., Tergny, J. and Laritchev, O. Linear programming with multiple objective functions: Step method (STEM), Mathematical Programming 1(3) (1971) 366-375.
2. Buchanan, J.T. A naive approach for solving MCDM problems: the GUESS method, Journal of the Operational Research Society 48 (1997)202-206.
3. Chankong, V. and Haimes, Y.Y. The interactive surrogate worth trade-off (ISWT) method for multiobjective decision-making, in: S. Zionts (Eds.), Multiple Criteria Problem Solving, Berlin: Springer, 1978, pp. 42-67.
4. Ehrgott, M. Multicriteria Optimization, Springer, Berlin, 2005.
5. Engau, A. and Wiecek, M.M. Generating E-efficient solutions in multiobjective programming, European Journal of Operational Research 177(2007) 1566-1579.
6. Eschenauer, H.A., Osyczka, A. and Schafer, E. Interactive multicriteria optimization in design process, in: Eschenauer, H., Koski, J., Osyczka A., (Eds.), Multicriteria Design Optimization Procedures and Applications, Berlin: Springer, 1990, pp. 71-114.
7. Gardiner, L. and Steuer, R.E. Unified interactive multiple objective programming, European Journal of Operational Research 74(3) (1994) 391-406.
8. Gardiner, L. and Steuer, R.E. Unified interactive multiple objective programming: an open architecture for accommodating new procedures, Journal of the Operational Research Society 45(12) (1994) 1456-1466.
9. Geoffrion, A. Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications 22 (1968) 618-630.
10. Ghaznavi-ghosoni, B.A. and Khorram, E. On approximating weakly/properly efficient solutions in multiobjective programming, Mathematical and Computer Modelling 54 (2011) 3172-3181.
11. Ghaznavi-ghosoni, B.A., Khorram, E. and Soleimani-damaneh, M. Scalarization for characterization of approximate strong/weak/proper effciency in multiobjective optimization, Optimization, 62 (6) (2013) 703-720.
12. Ghaznavi, M. Optimality conditions via scalarization for approximate quasi efficiency in multiobjective optimization, Filomat, accepted.
13. Hosseinzadeh Lotfi, F., Jahanshahloo, G.R., Ebrahimnejad, A., Soltanifar, M. and Mansourzadeh, S.M. Target setting in the general combinedoriented CCR model using an interactive MOLP method, Journal of Computational and Applied Mathematics 234(1) (2010) 1-9.
14. Jahn, J. Vector Optimization: Theory, Applications, and Extensions, Springer-Verlag, Berlin, Germany, 2004.
15. Kaliszewski, I. A modified weighted Tchebycheff metric for multiple objective programming, Computers and Operations Research 14 (1987) 315-323.
16. Kaliszewski, I. A theorem on nonconvex functions and its application to vector optimization, European Journal of Operational Research 80 (1995)439-449.
17. Kaliszewski, I. Using trade-off information in decision making algorithms, Computers and Operations Research 27 (2000) 161-182.
18. Kaliszewski, I. and Michalowski, W. Efficient solutions and bounds on tradeoffs, Journal of Optimization Theory and Applications 94 (1997) 381-394.
19. Kaliszewski, I., Miroforidis, J. and Podkopaev, D. Interactive multiple criteria decision making based on preference driven evolutionary multiobjective optimization with controllable accuracy, European Journal of Operational Research 216 (2012) 188-199.
20. Korhonen, P. Reference direction approach to multiple objective linear programming: Historical overview, in: Karwan, M. H., Spronk,J., Wallenius, J., (Eds.), Essays in Decision Making: A Volume in Honour of Stanley Zionts, Springer-Verlag, Berlin, Heidelberg, 1997, pp. 74-92.
21. Luque, M., Ruiz, F. and Miettinen, K. Global formulation for interactive multiobjective optimization, OR Spectrum 33(1) (2011) 27-48.
22. Luque, M., Yang, J.B. andWong, B.Y.H. PROJECT method for multiobjective optimization based on the gradient projection and reference point, IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans 39(4) (2009) 864-879.
23. Miettinen, K. Nonlinear Multiobjective Optimization, Kluwer Academic Publishers, Dordrecht, 1999.
24. Nakayama, H. and Sawaragi, Y. Satisficing trade-off method for multiobjective programming, in: M. Grauer, A.P. Wierzbick (Eds.), Interactive Decision Analysis, Berlin: Springer, 1984, pp. 113-122.
25. Narula, S.C., Kirilov, L. and Vassilev, V. Reference direction approach for solving multiple objective nonlinear programming problems, IEEE Transactions on Systems, Man, and Cybernetics 24 (1994) 804-806.
26. Park, K.S. and Shin, D.E. Interactive multiobjective optimization approach to the inputoutput design of opening new branches, European Journal of Operational Research 220 (2012) 530-538.
27. Romero, C. Extended lexicographic goal programming: a unified approach, Omega 29 (2001) 63-71.
28. Ruiz, F., Luque, M. and Miettinen, K. Improving the computational efficiency in a global formulation (GLIDE) for interactive multiobjective optimization, Annals of Operations Research 197(1) (2012) 47-70.
29. Sakawa, M. Interactive multiobjective decision making by the sequential proxy optimization technique: SPOT, European Journal of Operational Research 9 (4) (1982) 386-396.
30. Taras, S. and Woinaroschy, A. An interactive multiobjective optimization framework for sustainable design of bioprocesses, Computers & Chemical Engineering 43 (2012) 10-22.
31. Vassilev, V., Narula, S.C. and Gouljashki, V.G. An interactive reference direction algorithm for solving multiobjective convex nonlinear integer programming problems, International Transactions in Operational Research 8(4) (2001) 367-380.
32. Vassileva, M., Miettinen, K. and Vassilev, V. Generalized scalarizing problem for multicriteria optimization, IIT Working Papers IIT/WP-205, Institute of Information Technologies, Bulgaria, 2005.
How to Cite
Ghaznavi mehrdad, IlatiM., & KhorramE. (2016). An interactive algorithm for solving multiobjective optimization problems based on a general scalarization technique. Iranian Journal of Numerical Analysis and Optimization, 6(1), 79-101. https://doi.org/10.22067/ijnao.v6i1.44631
Section
Research Article