25 Innovations and Info-Gaps: An Overview 269 Fig. 25.6 Robustness and opportuneness curves for innovative (dashed) and state of the art (solid) options The innovation dilemma is manifested graphically by the intersection of the robustness curves in Fig. 25.5, and this intersection is the basis for the resolution. Innovative option 1 is more robust than option 2 for highly demanding requirements, xc > 2.4, and hence option 1 is preferred for this range of performance requirements. Likewise, state-of-theart option 2 is more robust for more modest requirements, xc < 2.4, and hence option 2 is preferred for this lower range of performance requirements. The robust-satisficing designer will be indifferent between the two options for performance requirements at or close to the intersection value of xc D2.4. Considerations other than robustness can then lead to a decision. Figure 25.6 shows the robustness curves from Fig. 25.5 together with the opportuneness curves, Eq. (25.7), for the same parameter values. We note that the innovative (dashed) option is more opportune (smaller b ˇ) than the state of the art (solid) for all values of xw exceeding the nominal innovative value. Designers tend to be risk averse and to prefer robust-satisficing over opportune windfalling. Nonetheless, opportuneness can “break the tie” when robustness does not differentiate between the options at the specified performance requirement. Functional uncertainty. We have discussed the info-gap robustness function and its properties of trade off, zeroing, and cost of robustness. We have illustrated how these concepts support the decision making process, especially when facing an innovation dilemma. Finally, we have described the info-gap opportuneness function and its complementarity to the robustness function. These ideas have all been illustrated in the context of a one-dimensional linear system with uncertainty in a single parameter. In most applications with severe uncertainty the info-gaps include multiple parameters as well as uncertainty in the shapes of functional relationships. We now extend the previous example to illustrate the modeling and management of functional uncertainty. This will also illustrate how uncertain probabilistic models can be incorporated into an info-gap robust-satisficing analysis. Let the stiffness coefficient, K in Eq. (25.2), be a random variable whose estimated probability density function (pdf), Qp.K/, is normal with mean and variance 2. We are confident that this estimate is accurate for Kwithin an interval around of known size ˙ıs. However, outside of this interval of Kvalues the fractional error of the pdf is unknown. In other words, we are highly uncertain about the shape of the pdf outside of the specified interval. This uncertainty derives from lack of data with extreme K values and absence of fundamental understanding that would dictate the shape of the pdf. We face “functional uncertainty” that can be represented by info-gap models of many sorts, depending on the type of information that is available. Given the knowledge available in this case we use the following fractional-error info-gap model: U.h/ Dnp.K/ W R1 1 p.K/ dKD1; p.K/ 0 for all K; p.K/ D Qp.K/ for jK j ıs; ˇ ˇ ˇ ˇ p.K/ Qp.K/ Qp.K/ ˇ ˇ ˇ ˇ h forjK j >ıso; h 0 (25.12) The first row of this info-gap model states that the set U(h) contains functions p(K) that are normalized and non-negative, namely, mathematically legitimate pdf’s. The second line states the these functions equal the estimated pdf in the specified interval around the mean, . The third line states that, outside of this interval, the functions inU(h) deviate fractionally from the estimated pdf by no more than h. We add the technical simplification that the pdf’s in U(h) are non-atomic: they contain no delta functions. In short, this info-gap model is the unbounded family of nested sets, U(h), of pdf’s that are known within the interval ˙ıs but whose shapes are highly uncertain beyond it. This is one example of an info-gap model for uncertainty in the shape of a function.
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