25 Innovations and Info-Gaps: An Overview 267 Achieving a windfall requires a favorable surprise, so the windfaller asks: what is the lowest horizon of uncertainty at which windfall is possible (though not necessarily guaranteed)? The answer is the opportuneness function, defined as: b ˇ.xw/ Dmin h W max K2U.h/ x xw (25.6) Reading this equation from left to right we see that the opportuneness, b ˇ, is the minimum horizon of uncertainty, h, up to which at least one realization of the uncertain stiffness Kin the uncertainty set U(h) results in displacement x at least as large as the wonderful windfall valuexw. The opportuneness function, b ˇ.xw/, is the complement of the robustness function b h.xc/ in Eq. (25.4). We see, for example, that the min and max operators in these two equations are reversed. This is the mathematical manifestation of the inverted meaning of these two functions. Robustness is the greatest uncertainty that guarantees the required outcome, while opportuneness is the lowest uncertainty that enables the aspired outcome. Opportuneness is useful for decision support because a more opportune option is better able to exploit propitious uncertainty than a less opportune option. An option whose b ˇ value is small is opportune because windfall can occur even at low horizon of uncertainty. The opportune windfaller prioritizes options according to the smallness of their opportuneness function values: an option with small b ˇis preferred over an option with large b ˇ. That is, “smaller is better” for opportuneness, unlike robustness for which “bigger is better”. Once again we note the logical inversion between robustness and opportuneness. Whether a decision maker prioritizes the options using robustness or opportuneness is a methodological decision that may depend on the degree of risk aversion of the decision maker. Furthermore, these methodologies may or may not prioritize the options in the same order. The opportuneness function is derived in a manner analogous to the derivation of Eq. (25.5), yielding: b ˇ.xw/ D 1 s Q K F xw (25.7) or zero if this is negative which occurs whenxw is so small, modest, and unambitious that it is possible even with the nominal design and does not depend on the potential for propitious surprise. The robustness and opportuneness functions, Eqs. (25.5) and (25.7), are plotted in Fig. 25.4. The opportuneness function displays zeroing and trade off properties whose meanings are the reverse of those for robustness. The opportuneness function equals zero at the nominal outcome, x DF=QK, like the robustness function. However, for the opportuneness function this means that favorable windfall surprise is not needed in order to enable the predicted outcome. The positive slope of the opportuneness function means that greater windfall (larger xw) is possible only at larger horizon of uncertainty. The robustness and opportuneness functions may respond differently to proposed changes in the design, as we now illustrate with Eqs. (25.5) and (25.7). From the first of these equations we note that b h decreases as the nominal stiffness, QK, increases. From the second equation we see that b ˇincreases as QKincreases: @ b h @QK <0; @ b ˇ @QK >0 (25.8) Fig. 25.4 Robustness and opportuneness curves of Eqs. (25.5) and (25.7). F/sD3. QK=s D1
RkJQdWJsaXNoZXIy MTMzNzEzMQ==