A general decomposition approach for multi-criteria decision trees

In this paper, we address the dynamic and multi-criteria decision-making problems under uncertainty, generally represented by multi-criteria decision trees. Decision-making consists of choosing, at each period, a decision that maximizes the decision-maker outcomes. These outcomes should often be measured against a set of heterogeneous and conflicting criteria. Generating the set of non-dominated solutions is a common approach considered in the literature to solve the multi-criteria decision trees, but it becomes very challenging for large problems. We propose a new approach to solve multi-criteria decision trees without generating the set of all non-dominated solutions, which should reduce the computation time and the cardinality of the solution set. In particular, the proposed approach combines the advantages of decomposition with the application of multi-criteria decision aid (MCDA) methods at each decision node. A generalization of the Bellman’s principle of decomposition to the multi-criteria context is put forward. A decomposition theorem is therefore proposed. Under the sufficient conditions stated by the theorem, the principle of decomposition will generate the set of best compromise strategies. Seven MCDA methods are then characterized (lexicographic, weighted sum, multi-attribute value theory, TOPSIS, ELECTRE III, and PROMETHEE II) against the conditions of the theorem of decomposition and against other properties (neutrality, anonymous, fidelity, dominance, independency), in order to confirm or infirm their applicability with the proposed decomposition principle. Moreover, the relationship between independency and temporal consistence is discussed as well as the effects of incomparableness, rank reversals, and use of thresholds. Two conjectures resulted from this characterization.