Gholam H. Askarirobati Akbar Hashemi Borzabadi Aghileh Heydari‎‎‎


Detecting the Pareto optimal solutions on the Pareto frontier is one of the most important topics in multiobjective optimal control problems. In real-world control systems, there is needed for the decision-maker to apply their own opinion to find the preferred solution from a large list of Pareto optimal solutions. This paper presents a class of axial preferred solutions for multiobjective optimal control problems in contexts in which partial information on preference weights of objectives is available. These solutions combine both the idea of improvement axis and Pareto optimality with respect to preference information. The axial preferred solution, in addition to taking considerations of decision-makers, provides continuous functions for control ling chemical processes. Numerical results are presented for two problems of chemical processes with two different preferential situations.

Article Details


Multiobjective optimal control;, Improvement axis;, Partial information;, Axial preferred solutions.

[1] Banihashemi, N. and Kaya, C.Y. Inexact restoration for Euler discretization of box-constrained optimal control problems, J. Optim. Theory Appl. 156 (2013), 726–760.
[2] Dontchev, A.L. and Hager, W.W. The Euler approximation in state con strained optimal control problems, Math. Comput. 70 (2001), 173–203.
[3] Dontchev, A.L., Hager, W.W. and Malanowski, K. Error bound for Euler approximation of a state and control constrained optimal control problem, Numer. Funct. Anal. Optim. 21 (6) (2000), 653–682.
[4] Fourer, R., Gay, D.M. and Kernighan, B.W. AMPL: A modeling languag for mathematical programming, 2nd ed. Brooks/Cole, New York 2003.
[5] Ghane-Kanafi, A. and Khorram, E. A new scalarization method for finding the efficient frontier in non-convex multi-objective problems, Appl. Math. Model. 39 (2015), 7483–7498.
[6] Hwang, C.L. and Masud, A.S.M. Multiple objective decision making, methods and applications: A state-of-the-art survey, Springer-Verlag, 1979.
[7] Ida, M. Multi-objective optimal control through linear programming with interval objective function, Proceedings of the 36th SICE Annual Confer ence, 1997.
[8] Jaimes, A.L., Santana Quintero, L.V. and Coello, C.C.A. Ranking methods in many-objective evolutionary algorithms, Nature-Inspired Algorithms for Optimisation. (2009), 413–434.
[9] Kalai, E. Proportional solutions to bargaining situations: Interpersonal utility comparisons, Econometrica. 45 (7) (1977), 1623–1630.
[10] Kaya, C.Y. Inexact restoration for Runge–Kutta discretization of optimal control problems, SIAM J. Numer. Anal. 48 (4) (2010), 1492–1517.
[11] Kaya, C.Y. and Martínez, J.M. Euler discretization for inexact restoration and optimal control, J. Optim. Theory Appl. 134 (2007), 191–206.
[12] Logist, F. and Van Impe, J. Multiple objective optimization of cyclic chemical systems with distributed parameters, Control. Appl. Optim. 7(2009), 295–300.
[13] Logist, F., Houska, B., Diehl, M. and Van Impe, J.F. Fast Pareto set generation for nonlinear optimal control problems with multiple objectives, Struct. Multidiscip. Optim. 42, (2010) 591–603.
[14] Logist, F., Vallerio, M., Houska, B., Diehl, M. and Van Impe, J. Multi objective optimal control of chemical processes using ACADO toolkit, Com put. Chem. Eng. 37 (2012) 191–199.
[15] Mality, K. and Maiti, M. Numerical approach of multi-objective optimal control problem in imprecise environment, Fuzzy Optim. Decis. Ma. 4(2005), 313–330.
[16] Ober-Blobaum, S., Ringkamp, M. and Zum, G. Felde solving multiobjective optimal control problems in space mission design using discrete me chanics and reference point techniques, 51st IEEE Conference on Decision
and Control, (2012), 5711–5716.
[17] Ohno, H., Nakanishi, E. and Takamatsu, T. Optimal control of a semi batch fermentation, Biotechnol. Bioeng. 18 (1976), 847–864
[18] Peitz, S. and Dellnitz, M, A survey of recent trends in multiobjective optimal control–surrogate models, feedback control and objective reduction, Math. Comput. Appl, (2018), 1–33.
[19] Severino Leal, U.A., Silva, G.N. and Lodwick, W. A. Multi-objective optimization in optimal control problem with interval-valued objective function, Proceeding Series of the Brazilian Society of Applied and Computational
Mathematics. 3 (1) (2015).
[20] Vassiliadis, V., Balsa-Canto, E. and Banga, J. Second-order sensitivities of general dynamic systems with application to optimal control problems, Chem. Eng. Sci. 54 (1999), 3851–3860.
[21] Wächter, A. and Biegler, L.T. On the implementation of a primal-dual interior point filter line search algorithm for large-scale nonlinear program ming, Math. Program. 106 (2006), 25–57.
[22] Zarei, H. and Rezai Bahrmand, M. Multi-objective optimal control of the linear wave equation, Ain Shams Eng. J. 5 (2014), 1299–1305.
[23] Zitzler, E., Laumanns, M. and Thiele, L. SPEA2: Improving the strength Pareto evolutionary algorithm, Zurich, Switzerland: Swiss Federal Institute Technology 2001.
How to Cite
AskarirobatiG. H., Hashemi BorzabadiA., & Heydari‎‎‎A. (2020). Axial preferred solutions for multiobjective optimal control problems: An application to chemical processes. Iranian Journal of Numerical Analysis and Optimization, 10(1), 19-32. https://doi.org/10.22067/ijnao.v10i1.75932
Research Article