Maciej Nowak , Tadeusz Trzaskalik

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In this paper we consider a multi-stage, multi-criteria discrete decision process under risk. We use a discrete, stochastic dynamic programming approach based on Bellman’s principle of optimality. We assume that the decision maker determines a quasi-hierarchy of the criteria considered; in other words, he or she is able to determine to what extent the optimal expected value of a higher-priority criterion can be made worse to improve the expected value of a lower-priority criterion. The process of obtaining the final solution can be interactive. Based on the observations of the consecutive solutions, the decision maker can modify the aspiration levels with respect to the criteria under consideration, finally achieving a solution which satisfies him/her best. The method is illustrated on an example based on fictitious data.


dynamic programming, decision making under risk, interactive approach, quasihierarchical method


Bakker B., Zivkovic Z., Krose B., (2005), Hierarchical Dynamic Programming for Robot Path Planning, in: Intelligent Robots and Systems 2005 (IROS 2005), 2756–2761.

Bellman R., (1957), Dynamic Programming, Princenton University Press.

Daellenbach H. G., De Kluyver C. A., (1980), Note on Multiple Objective Dynamic Programming, Journal of the Operational Research Society, 31 (7), 591–594.

Dempster M. A. H., (2006), Sequential Importance Sampling Algorithms for Dynamic Stochastic Programming, Journal of Mathematical Sciences, 133 (4), 1422–1444.

Elmaghraby S. E., (1970), The Theory of Networks and Management Science, part I, Management Science, 17 (1), 1–34.

Hatzakis I., Wallace D., (2006), Dynamic Multi-Objective Optimization with Evolutionary Algorithms: a Forward-looking Approach, in: Proceedings of the 8th Annual Conference on Genetic and Evolutionary Computation, 1201–1208.

Nowak M., (2014), Quasi-Hierarchical Approach in Multiobjective Decision Tree, Studia Ekonomiczne Uniwersytetu Ekonomicznego w Katowicach, 208, 59–73 (in Polish).

Nowak M., Trzaskalik T., (2017), Optimal and Near Optiomal Strategies in Discrete Stochastic Multiobjective Quasi-Hierarcical Dynamic Poblems, in: Doerner K. F., Ljubic I., Pflug G., Tragler G., (eds.), Operations Research Proceedings, Springer, 295–300.

Sethi S. P., Yan H., Zhang H., Zhang Q., (2002), Optimal and Hierarchical Controls in Dynamic Stochastic Manufacturing Systems: A Survey. Manufacturing & Service Operations Management, 4 (2), 133–170.

Sethi S., Zhang Q., (1994), Hierarchical Production Planning in Dynamic Stochastic Manufacturing Systems: Asymptotic Optimality and Error Bounds, Journal of Mathematical Analysis and Applications, 181 (2), 285–319.

Shapiro A., (2012), Minimax and Risk Averse Multistage Stochastic Programming, European Journal of Operational Research, 219 (3), 719–726.

Tempelmeier H., Hilger T., (2015), Linear Programming Models for a Stochastic Dynamic Capacitated Lot Sizing Problem, Computers & Operations Research, 59, 119–125.

Topaloglou N., Vladimirou H., Zenios S. A., (2008), A Dynamic Stochastic Programming Model for International Portfolio Management, European Journal of Operational Research, 185 (3), 1501–1524.

Trzaskalik T., (1990), Multiobjective Discrete Dynamic Programming. Theory and Applications in Economics, The Karol Adamiecki University of Economics in Katowice Press, Katowice (in Polish).

Trzaskalik T., (1998), Multiobjective Analysis in Dynamic Environmnent, The Karol Adamiecki University of Economics in Katowice Press, Katowice, 1998.

Woerner S., Laumanns M., Zenklusen R., Fertis A., (2015), Approximate Dynamic Programming for Stochastic Linear Control Problems on Compact State Spaces, European Journal of Operational Research, 241 (1), 85–98.

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