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thethreshold according to individual concerns,selec-tion will often be driven by some feature of the prob-lem under consideration.For example,the decisionmaker may be concerned with some project causingextreme costs,e.g.,a budget overrun,and may thenbe chosen to equal the budgeted amount—see the ex-
ample in Sections 7 and 8.Some physical constraint(e.g.,height of a levee)or a date in time(scheduleoverrun)may be other factors influencing selectionof.The use of another conditional expected value,conditioned on the outcome falling in the upper 100(1)percent tail of the cumulative probability dis-tribution of outcomes(rare event),will be addressed inthe companion paper in this issue(Frohwein et al.2000).The use of conditional expected values as a mea-sure of the risk of extreme events in MODT analysispresents a challenge because conditional expected
values cannot be optimized by the well-known method(Raiffa 1968,Clemen 1996)of averaging out andfolding back(Haimes et al.1990,Haimes 1998).This
paper develops an approach for sequential optimiza-
tion of conditional expected values in MODT analy-
sis,conditioned on the magnitude of the outcome ex-
ceeding a threshold.
The conditional expected value considered in
this paper does not account for the probability of ex-
periencing an extreme event with a magnitude of at
least.It may be argued that a decision maker would
be interested not only in knowing(and minimizing)
the conditional expected outcome,given that an ex-
treme event occurs,but also in knowing the probabil-
ity of such an event occurring.However,no single
measure can capture all the facets of the risk of ex-
treme events,and a decision maker concerned with
the risk of extreme events could indeed be interested
in the conditional expected outcome.This informa-
tion will typically be used in conjunction with other
data,among them the probability of experiencing an
extreme event.Thus,although it is not claimed that
the conditional expected value is the sole and com-
prehensive measure of the risk of extreme events,it
does offer insight on one facet of risk.For the use of
conditional expected values as a measure of the risk
of extreme events,see for example,Asbeck and
Haimes(1984).Glickman and Sherali(1991),Karls-
son and Haimes(1988,1989),Lambert et al.(1994),
Mitsiopoulos et al.(1991),Sherali et al.(1997),and Si-
vakumar et al.(1993,1995).Erkut(1995)and Thomp-
son et al.(1997)voice some critiques on the use of
conditional expected values as objective functions.
The paper is organized as follows.First,differ-
ences between SODTs and MODTs are highlighted.
Then,the obstacle to averaging out and folding back
conditional expected values in decision trees is dis-
cussed.Next,the central insight from multiobjective
optimization that enables folding back and averaging
out of the risk of severe events is developed.It is seen
that some policies can be eliminated at intermediate
nodes of the decision tree.In the subsequent section,
the idea is extended to multiple objectives.The opti-
mization process is summarized and depicted in a
flowchart.Then,an example is provided,followed by
some concluding remarks.
2.BACKGROUND:SINGLE-VS.
MULTIPLE-OBJECTIVE DECISION TREES
The reader is assumed to be familiar with SODTs
and the folding-back-and-averaging-out procedure
typically used to solve them.Averaging o