January 2020
(the output; Corrente, Figueira, & Greco, 2014; Hyde & Maier, 2006; Hyde, Maier, & Colby, 2004; Rocco & Tarantola, 2014; Yu, Guikema, Briaud, & Burnett, 2012). The approaches, based onMonte Carlo simulation, consider each uncertain factor as a randomvariable with known probability density functions. As a result, the AQ of each alternative, and therefore its ranking, also become randomvariables, with approximated probability distributions. In such situations, the decisionmaker could performprobability distribution evaluations. For example, the decision maker could be interested in determining not only what the worst rank of a specifc alternative is, but also its probability and volatility (risk evaluation). In the standard approach, the probability of an alternative being ranked as in the BCR is selected as the synthetic attribute probability able to characterize the alternatives under uncertainty. The stochastic nature of the AQ of each alternative could be further assessed in order to refect the risk evaluation induced by uncertainty. In this case, it is required to compare several random variables synthesized through their percentiles and statistical moments. Several approaches have been proposed to this end, such as a simple comparison of the expected value, the expected utility (Von Neumann&Morgenstern, 1947), the use of loworder moments (Markowitz, 1952), risk measures (Jorion, 2007; Mansini, Ogryczak, & Speranza, 2007; Rockafellar &Uryasev, 2000), the PartitionedMultiobjective RiskMethod (Asbeck & Haimes, 1984; Haimes 2009), and the stochastic dominance theory (Levy, 2006), among others. Therefore, the final assessment is derived using a combined approach based on a nonparametric aggregation rule (using the concept of average rank) for attributes 1 and 2; a simple procedure for score assignment for attribute 3; and a lexicographic rule . In addition, a preliminary analysis of the alternatives is performed by using a Hasse diagram (Bruggemann & Patil, 2011). To the best of the researcher’s knowledge, this type of combined assessment has not been reported in the literature. Average Rank Approach Let P defne a set of n objects (e.g., alternatives) to be analyzed and let the descriptors q 1 , q 2 ..., q m defne m diferent attributes or criteria selected to assess the objects in P (e.g., cost, availability, environmental impact). It is important that attributes are defned to refect, for example, that a low value indicates low rankings (best positions), while a high value indicates high ranking (worst positions; Restrepo, Brüggemann, Weckert, Gerstmann, & Frank, 2008). However, for a given problem or case study, this convention could be reversed.
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Defense ARJ, January 2020, Vol. 27No. 1 : 60-107
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