Engineering Design under Imprecise Probabilities: Computational Complexity
In engineering design problems, we want to make sure that a certain quantity c of the designed system lies within given bounds - or at least that the probability of this quantity to be outside these bounds does not exceed a given threshold. We may have several such requirements - thus the requiremen...
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| Lenguaje: | English |
| Publicado: |
Universidad de La Frontera. Departamento de Matemática y Estadística.
2011
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| Acceso en línea: | http://www.scielo.cl/scielo.php?script=sci_arttext&pid=S0719-06462011000100007 |
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| Sumario: | In engineering design problems, we want to make sure that a certain quantity c of the designed system lies within given bounds - or at least that the probability of this quantity to be outside these bounds does not exceed a given threshold. We may have several such requirements - thus the requirement can be formulated as bounds [Fc(x), Fc(x)] on the cumulative distribution function Fc(x) of the quantity c; such bounds are known as a p-box. The value of the desired quantity c depends on the design parameters a and the parameters b characterizing the environment: c = f(a, b). To achieve the design goal, we need to find the design parameters a for which the distribution Fc(x) for c = f(a, b) is within the given bounds for all possible values of the environmental variables b. The problem of computing such a is called backcalculation. For b, we also have ranges with different probabilities - i.e., also a p-box. Thus, we have backcalculation problem for p-boxes. For p-boxes, there exist efficient algorithms for finding a design a that satisfies the given constraints. The next natural question is to find a design that satisfies additional general, the problem of finding such a design is computationally difficult (NP-hard). We show that this problem is NP-hard already in the simplest possible linearized case, when the dependence c = f(a, b) is linear. We also provide an example when an efficient algorithm is possible. |
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