تعیین جواب‌های تقریباً کارای مسائل بهینه‌سازی چندهدفه با استفاده از روش اسکالرسازی مقید ترکیبی
محورهای موضوعی : آمار
مهرداد غزنوی
1
(گروه ریاضی کاربردی و علوم کامپیوتر، دانشکده علوم ریاضی، دانشگاه صنعتی شاهرود، شاهرود، ایران)
فرشته اکبری
2
(‏دانشکده ریاضی و علوم کامپیوتر، دانشگاه صنعتی امیرکبیر، تهران، ایران)
اسماعیل خرم
3
(دانشکده علوم ریاضی و کامپیوتر، دانشگاه صنعتی امیر کبیر، تهران، ایران)
کلید واژه: Scalarization method. Approximate efficient solutions, Approximate optimality. , Multiobjective optimization, Proper efficiency,
چکیده مقاله :
در این مقاله، جواب‌های تقریباً کارای ( -کارای) مسائل بهینه‌سازی چندهدفه مورد بررسی قرار می‌گیرند. یک دسته از مهم ترین روش‌ها برای حل مسائل چندهدفه، استفاده از تکنیک‌های اسکالرسازی است. در این روش‌ها یک مسأله تک‌هدفه متناظر با مسأله چندهدفه حل می‌شود و ارتباط بین جواب‌های بهینه مسأله‌ی تک‌هدفه و جواب‌های کارای (سره، ضعیف) مسأله‌ی چندهدفه بررسی می‌شود. در این مقاله، ترکیبی از روش‌های اسکالرسازی مقید اصلاح شده (modified constrained) و مقید انعطاف‌پذیر (elastic constrained) در نظر گرفته می‌شود و با کمک آن شرایطی لازم و کافی برای تولید جواب‌های تقریباً کارا (ضعیف، سره) ارائه خواهد شد. نتایج بدست آمده را با شرایط لازم و کافی حاصل از روش‌های مقید اصلاح شده و مقید انعطاف‌پذیر مقایسه می‌کنیم. قضایای ارائه شده بدون هیچ شرط تحدبی برای هر یک از توابع هدف در مسأله‌ی بهینه‌سازی چندهدفه برقرار هستند. برخلاف بسیاری از روش‌های قبلی، نتایج بدست آمده برای مسائل چندهدفه با فضای هدف بیکران نیز برقرار هستند.
In this paper, approximate efficient ( -efficient) solutions of multiobjective optimization problems are investigated. One of the most important methods for solving multiobjective optimization problems is to use scalarization techniques. In these methods, a single objective optimization problem corresponding to the multiobjective problem is solved, and the relationship between optimal solutions of the single objective problem and (weakly, properly) efficient solutions of the multiobjective problem is investigated. In this paper, a combination of the modified constrained and elastic constrained scalarization methods is considered, which will provide necessary and sufficient conditions for generating approximate (weakly, properly) efficient solutions. We compare the results with the necessary and sufficient conditions obtained from the modified constrained and the elastic constrained methods. The presented results can be applied for every multiobjective optimization problem without any convexity assumption for the objective functions. ‎Unlike many of the previous methods, the obtained results are also consistent with multiobjective problems with unbounded criterion space.
[1] Shao, L., Ehrgott, M. (2008). Approximately solving multiobjective linear programmes in objective space and an application in radiotherapy treatment planning, Mathematical Methods of Operations Research, 68, 257-276.
[2] Shao, L., Ehrgott, M. (2008). Approximating the nondominated set of an MOLP by approximately solving its dual problem, Mathematical Methods of Operations Research, 68, 469–492.
[3] Ehrgott, M., Holder, A. (2017). Operations research methods for optimization in radiation oncology, Journal of Radiation Oncology Informatics, 6(1), 1-41
[4] Drake, J.H., Starkey, A., Owusu, G., Burke, E.K. (2020). Multiobjective evolutionary algorithms for strategic deployment of resources in operational units, European Journal of Operational Research, 282(2), 729-740.
[5] Eichfelder, G. (2008). Adaptive Scalarization Methods in Multiobjective Optimization. Springer-Verlag, Berlin, Heidelberg.
[6] Soylu, B., Katip, K. (2019). A multiobjective hub-airport location problem for an airline network design, European Journal of Operational Research, 277(2), 412-425.
[7] Soleimani-damaneh, M. (2011). An optimization modelling for string selection in molecular biology using Pareto optimality, Applied Mathematical Modelling, 35, 3887-3892.
[8] Soleimani-damaneh, M. (2011). On some multiobjective optimization problems arising in biology, International Journal of Computer Mathematics, 88, 1103–1119.
[9] Shi, J., Song, J., Song, B., Lu, W.F. (2019). Multi-objective optimization design through machine learning for drop-on-demand bioprinting, Engineering, 5(3), 586-593.
[10] Edgeworth, F.Y. (1881). Mathematical Psychics, Kegan Paul, London.
[11] Pareto, V. (1964). Course dEconomics Politique, Libraire Droze, Geven.
[12] Ehrgott, M. (2005). Multicriteria Optimization, Berlin: Springer.
[13] Jahn, J. (2004). Vector Optimization: Theory, Applications, and Extensions, Springer-Verlag, Berlin, Germany.
[14] Miettinen, K. (1999). Nonlinear Multiobjective Optimization, Kluwer Academic Pablishers, Dordrecht.
[15] Akbari, F., Ghaznavi, M., Khorram, E. (2018). A revised Pascoletti-Serafini scalarization method form multiobjective optimization problems, Journal of Optimization Theory and Applications, 178(2), 560-590.
[16] Ghaznavi, M., Akbari, F., Khorram, E. (2019). Optimality conditions via a unified direction approach for (approximate) efficiency in multiobjective optimization, Optimization Methods and Software, In press, 1-26.
[17] Kutateladze, S.S. (1979). Convex ε-programming, Soviet Mathematics-Doklady, 20, 391–393.
[18] Loridan, P. (1984). ε-solutions in vector minimization problems, Journal of Optimization Theory and Applications, 43 (2), 265–276.
[19] White, D.J. (1986). Epsilon efficiency, Journal of Optimization Theory and Applications, 49 (2), 319–337.
[20] Engau, A. and Wiecek, M.M. (2007). Generating epsilon-efficient solutions in multi-objective programming, European Journal of Operational Research, 177, 1566–1579.
[21] Gutierrez, C., Jimenez, B., Novo,V. (2006). A unified approach and optimality conditions for approximate solutions of vector optimization problems, SIAM Journal on Optimization, 17, 688-710.
[22] Gutierrez, C., Jimenez, B., Novo,V. (2010). Optimality conditions via scalarization for a new ε-efficiency concept in vector optimization problems, European Journal of Operational Research, 201, 11-22.
[23] Chuong, T.D., Kim, D.S. (2015). Approximate solutions of multiobjective optimization problems, Positivity, 20(1) (2015) 187-207.
[24] Fakhar, M., Mahyarinia, M.R., Zafarani, J. (2019). On approximate solutions for nonsmooth robust multiobjective optimization problems, Optimization, 68, 1653-1683.
[25] Hong, Z., Piao, G., Kim, D.S. (2019). On approximate solutions of nondifferentiable vector optimization problems with cone-convex objectives, Optimization Letters, 13, 891–906.
[26] Kim, D.S., Son, T.Q. (2018). An Approach to 𝜖-duality Theorems for Nonconvex Semi-infinite Multiobjective optimization Problems, Taiwanese Journal of Mathematics, 22(5), 1261-1287.
[27] Li, Z., Wang, S. (1998). ε-approximation solutions in multiobjective optimization, Optimization, 44, 161–174.
[28] Liu, J.C. (1999). ε-properly efficient solutions to nondifferentiable multi-objective programming problems, Applied Mathematics Letters, 12, 109–113.
[29] Ghaznavi-ghosoni, B.A., Khorram, E. (2011). On approximating weakly/properly efficient solutions in multi-objective programming, Mathematical and Computer Modelling, 54 (11), 3172-3181.
[30] Ghaznavi-ghosoni, B.A., Khorram, E., Soleimani-damaneh, M. (2013). Scalarization for characterization of approximate strong/weak/proper efficiency in multi-objective optimization, Optimization, 62 (6), 703-720.
[31] Khaledian, K., Khorram, E., Karimi, B. (2014). Characterizing ε-Properly Efficient Solutions, Optimization Methods and Software, 30, 583 – 593.
[32] Rastegar, N., Khorram, E. (2014). A combined scalarizing method for multiobjective programming problems, European Journal of Operational Research, 236(1), 229-237.
[33] Ghaznavi, M. (2017). Optimality conditions via scalarization for approximate quasi efficiency in multiobjective optimization, Filomat, 31 (3) (2017) 671-680.
[34] Gao, Y., Xu, Z. (2019). Approximate proper efficiency for multiobjective optimization problems, Filomat, 33(18), 6091–6101.
[35] Geoffrion, A. (1968). Proper efficiency and the theory of vector maximization, Journal of Mathematical Analysis and Applications, 22(3) (1968) 618-630.
[36] Ehrgott, M., Ruzika, S. (2008). Improved ε-constraint method for multi-objective programming, Journal of Optimization Theory and Applications, 138, 375–396.
