268 Y. Ben-Haim and S. Cogan Recall that “bigger is better” for robustness while “smaller is better” for opportuneness. We see that any increase in QK will make both robustness and opportuneness worse, and any decrease in QKwill improve them both. We summarize this by saying the robustness and opportuneness are sympathetic with respect to change in stiffness. Now consider the estimated error, s in the info-gap model of Eq. (25.3). A smaller value of s implies greater confidence in the estimate, QK, while a larger s implies a greater propensity for error in the estimate. From Eq. (25.5) we see that the robustness improves ( b h increases) as s decreases: better estimate of QK implies greater robustness against uncertainty in K. In contrast, from Eq. (25.7) we see that the opportuneness gets worse ( b ˇ increases) as s decreases: lower opportunity for windfall as uncertainty of the estimate declines. In short: @ b h @s <0; @ b ˇ @s <0 (25.9) A change in the estimated error acts differently on robustness and opportuneness: by reducing the error of the estimated stiffness one increases the robustness but diminishes the opportuneness; increasing the error acts in the reverse. In short, robustness and opportuneness are antagonistic with respect to error in the estimated stiffness. An innovation dilemma occurs when the decision maker must choose between two options, where one is putatively better but more uncertain than the other. Technological innovations provide the paradigm for this dilemma. An innovation is supposedly better than the current state of the art, but the innovation is new so there is less experience with it and its behavior in practice may turn out worse than the current state of the art. We will illustrate an innovation dilemma with the previous example, demonstrating its resolution using the robustness functions of the two options. Consider two alternative designs of the linear elastic system, Eq. (25.2), one of which has lower estimated stiffness than the other: QK1 < QK2 (25.10) Both designs will operate under the same positive force, F, so the predicted displacement, x D F=QK, is greater with option 1. Thus option 1 is preferred based on the estimated stiffnesses and the requirement for large displacement. However, the putatively better option 1 is based on innovations for which the actual stiffness, in operation, is more uncertain than for option 2 which is the state of the art. Referring to the uncertainty estimate, s, in the info-gap model of Eq. (25.3), we express this as: s1 >s2 (25.11) The dilemma is that option 1 is putatively better, Eq. (25.10), but more uncertain, Eq. (25.11). This dilemma is manifested in the robustness functions for the two options, which also leads to a resolution, as we now explain. To illustrate the analysis we evaluate the robustness function for each option, Eq. (25.5), with the following parameter values: FD1, QK1 D1=6, s1 D1, QK2 D1=3 Ds2. The robustness curves are shown in Fig. 25.5. The innovative option 1 in Fig. 25.5 (dashed curve) is putatively better than the state of the art option 2 (solid curve) because the predicted displacement of option 1 is F=QK1 D6while the predicted displacement for option 2 is only 3. However, the greater uncertainty of option 1 causes a stronger trade off between robustness and performance than for option 2. The cost of robustness is greater for option 1 for xc values exceeding about 2, causing the robustness curves cross one another at about xc D2.4. Fig. 25.5 Robustness curves for innovative and state of the art options
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