A methodology for probabilistic aircraft technology assessment and selection under uncertainty

The high degree of complexity and uncertainty associated with aerospace engineering applications has driven designers and engineers towards the use of probabilistic and statistical analysis tools in order to understand and design for that uncertainty. As a result, probabilistic methods have permeated the aerospace field to the extent that single point deterministic designs are no longer credible, particularly in systems analysis, performance assessment, technology impact quantification, etc. However, as statistics theory is not the primary focus of most aerospace practitioners, incorrect assumptions and flawed methods are often unknowingly used in design. A common assumption of probabilistic assessments in the field of aerospace is the independence of random variables. These random variables represent design variables, noise variables, technology impacts, etc., which can be difficult to correlate but do have underlying relationships. The justification for the assumed independence is usually not discussed in the literature even though this can have a substantial effect on probabilistic assessment and uncertainty quantification results. In other cases the dependence between random variables is acknowledged but intentionally ignored on the basis of difficulty in characterizing underlying random variable relationships, a strong bias towards methodological simplicity and low computational expense, and the expectation of modest strength in random variable dependence. Probabilistic assessments also yield large amounts of data which is not effectively used due to the sheer volume of data and poor traceability to the drivers of uncertainty. The literature shows optimization techniques are resorted to in order to select from competing alternatives in multiobjective spaces, however, these techniques generally do not handle uncertainty well. The motivating question is, how can improvements be made to the probabilistic assessment process for aircraft technology assessments that capture technology impact tradeoffs and dependencies, and ultimately enable decision makers to make an axiomatic and rational selection under uncertainty? This question leads to the research objective of this work which is to develop a methodology ``to quantify and characterize aviation's environmental impact, uncertainties, and the trade-offs and interdependencies among various impacts'' \cite{Council2010}, in order to assess and select future aircraft technologies. Copula theory is suggested to address the problem of assumed independence on the input side of probabilistic assessments in aerospace applications. Copulas are functions that can be used to define probabilistic relationships between random variables. They are well documented in the literature and have been used in many fields such as the statistics, finance, and insurance industries. They can be used to quantify complex relationships, even if that is only qualitatively or notionally understood. In this way a designer's knowledge regarding uncertainty can be better represented and propagated to system level metrics through the probabilistic assessment. Utility theory is proposed as a solution to the challenge of effectively using output data from probabilistic assessments. Utility theory is a powerful tool used in economics, marketing, psychiatry, etc., to express preferences among competing alternatives. Utility theory can provide combined valuation to each alternative in a multiobjective design space while incorporating the uncertainty associated with each alternative. This can enable designers to rationally and axiomatically make selections consistent with their preferences, between complex solutions with varying degrees of uncertainty. This work provides an introduction to copula and utility theories for the aerospace audience. It also demonstrates how these theories can be applied in canonical problems to bridge gaps currently found in the literature with regards to probabilistic assessments of aircraft technologies. The key contributions of this research are (1) an Archimedean copula selection tree enabling practitioners to rapidly translate their qualitative understanding of dependence into copula families that can represent it quantitatively (2) estimation of the quantified effect of using copulas to capture probabilistic dependence in three representative aerospace applications (3) an expected utility formulation for axiomatically ranking and selecting aircraft technology packages under uncertainty and (4) a strategic elicitation procedure for multiattribute utility functions that does not need assumptions of independence conditions on preferences between the attributes. The proposed FAAST methodology is shown as an encompassing framework for the aircraft technology assessment and selection problem that fills capability gaps from the literature and supports the decision maker in a rational and axiomatic manner.Ph.D