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1.	Danielson, Mats, et al. (author) Computing Upper and Lower Bounds in Interval Decision Trees 2007 In: European Journal of Operational Research. - : Elsevier BV. - 0377-2217 .- 1872-6860. ; 181:2, s. 808-816 Journal article (peer-reviewed)abstract This article presents algorithms for computing optima in decision trees with imprecise probabilities and utilities. In tree models involving uncertainty expressed as intervals and/or relations, it is necessary for the evaluation to compute the upper and lower bounds of the expected values. Already in its simplest form, computing a maximum of expectancies leads to quadratic programming (QP) problems. Unfortunately, standard optimization methods based on QP (and BLP - bilinear programming) are too slow for the evaluation of decision trees in computer tools with interactive response times. Needless to say, the problems with computational complexity are even more emphasized in multi-linear programming (MLP) problems arising from multi-level decision trees. Since standard techniques are not particularly useful for these purposes, other, non-standard algorithms must be used. The algorithms presented here enable user interaction in decision tools and are equally applicable to all multi-linear programming problems sharing the same structure as a decision tree.

Danielson, Mats, et al. (author)
Computing Upper and Lower Bounds in Interval Decision Trees
2007
In: European Journal of Operational Research. - : Elsevier BV. - 0377-2217 .- 1872-6860. ; 181:2, s. 808-816
Journal article (peer-reviewed)abstract
- This article presents algorithms for computing optima in decision trees with imprecise probabilities and utilities. In tree models involving uncertainty expressed as intervals and/or relations, it is necessary for the evaluation to compute the upper and lower bounds of the expected values. Already in its simplest form, computing a maximum of expectancies leads to quadratic programming (QP) problems. Unfortunately, standard optimization methods based on QP (and BLP - bilinear programming) are too slow for the evaluation of decision trees in computer tools with interactive response times. Needless to say, the problems with computational complexity are even more emphasized in multi-linear programming (MLP) problems arising from multi-level decision trees. Since standard techniques are not particularly useful for these purposes, other, non-standard algorithms must be used. The algorithms presented here enable user interaction in decision tools and are equally applicable to all multi-linear programming problems sharing the same structure as a decision tree.

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