From the analysis of the extreme realisations with both parametric and non-parametric approaches, it is possible to conclude that
– When the response is monotonic with respect to a p-box, the extreme realisations are distributed as the bounding CDFs of that p-box;
– When the response is not monotonic with respect to a p-box, the distribution function of the extreme realisations is enclosed in the bounding CDFs of that p-box and may have a complicated form;
– In general, the two reconstructed CDFs of the extreme realisations are not dis- tributed as the parental model of probability;
– When the response is monotonic with respect to all p-boxes and when, for every p-box, the bounding CDFs are made of only two distribution functions (such as in the Beta model), the solution from the two approaches coincides.
5.4
Chapter summary
In this chapter, a novel Advanced Sampling strategy based on Line Sampling has been combined with the Random Set approach. The strategy has been tested on a simple numerical example, and further investigations on real engineering applications will be presented in the final chapter. The presented approach has shown not only to be very efficient, but also flexible as it can be adapted to any Advanced Sampling technique. The approach presented in this chapter and in Chapter 4, gained the IJAR silver award of ISIPTA15, targeted at young researchers who have demonstrated excellence in research
on imprecise probabilities. The results have been published in peer-reviewed conference papers (see, for example conference papers 3, 9 and 10 from the List of Publications), and have been presented at the 12th International Conference on Applications of Statistics and Probability in Civil Engineering (ICASP12), in July 2015.
Chapter 6
Forced Monte Carlo Strategy for
the Robust Scheduling of Multiple
Inspections with Random Sets
Advanced sampling techniques have demonstrated to be powerful tools to solve reli- ability problems with Random Sets. Another challenging problem in structural and system reliability, to be tackled with generalised uncertainty models, is the time de- pendent reliability problem. Every structure and system deteriorates with time and it is paramount to be able to assess the reliability at any given time. This extends to the challenging task of designing the optimal inspection schedule looking at the most economical solution, which can be found balancing the maintenance costs against the risk of failure.
The design of inspection schedules is a complex optimisation problem that requires the assessment of the system reliability. The solution to this problem can be found balancing the costs associated to inspection/repair activities against the benefits related to the faultless operation of the infrastructure. The objective is minimizing the total cost, obtained as the combination of maintenance costs and failure costs, while tuning some parameters, such as the number, time and quality of inspections.
The optimisation problem is formulated as a time-dependent reliability-based opti- misation problem, where both objective and constraint functions require the evaluation of upper and lower reliability bounds. The solution to this problem represents a real technological challenge, as the reliability assessment by means of Random Set is a computationally intensive task, which may take up to a few days to be completed on common processing units. In this chapter, an efficient and generally applicable and efficient numerical technique is proposed. The technique, integrated in OpenCossan, combines an Importance Sampling method derived from the concept of forced Monte Carlo simulation, with an optimisation strategy, which makes the interval reliability estimation particularly efficient.
6.1
Introduction
Preventive maintenance practice can be extremely cost-effective for mitigating dam- age accumulation of civil infrastructures. In fact, inspection and repair activities may prevent loss of serviceability or even partial collapse. However, making decisions as to whether and when to perform inspections is a very complex task, especially on real-scale engineering systems [83]. In fact, the realistic quantification of costs associated with inspections, repair and failure (i.e. loss of serviceability), requires the explicit consid- eration of the unavoidable uncertainties arising from the damage-propagation process, and from the inspection and repair activities.
Uncertainties may come from the inherent variability of the damage-propagation process or from the lack of available knowledge about the process itself. Random Sets are used as a comprehensive means of representing such heterogeneous uncertainties. Such an uncertainty model is quite general, and permits to assess the reliability and sensitivity of the computational model with respect to the uncertainty. In other words, the use of Random Sets adds robustness to the reliability analysis, making the analyst more aware of the effects of the uncertainties on the model response.
Reliability-based optimisation methods and techniques, as described e.g. in [32], are invoked to solve the problem.
Due to the explicit consideration of uncertainties, the design of maintenance activ- ities is an optimisation task that requires the assessment of reliability where number, times, and quality of inspections are the design variables and the total cost is the ob- jective function. For the formulation and solution to time-dependent reliability-based optimisation problems see e.g. [55] and [70]. The assessment of reliability both in the objective and in the constraints functions, and the consideration of multiple inspections, make this a stochastic discrete optimisation problem, which can be quite challenging to solve [75].
In this chapter, a general methodology for the efficient solution of the time-variant reliability-based maintenance optimisation problem is proposed, which is applicable to any case where the damage propagation law is known as input-output relationship. No restrictions in terms of number of inspections and number of uncertain parameters can be found. The methodology is derived from the concept of forced Monte Carlo simulation, used to evaluate the availability of plants [82], and it is exploited to efficiently assess the time-variant reliability conditional to the inspection outcomes, requiring only the execution of computationally inexpensive functions.
Here, Genetic Algorithms are used to drive the global optimisation, as the cost and constraint functions are stochastic and therefore, no information about the derivatives can be efficiently used to converge to the minimum. This comes with quite some ad- ditional numerical burden, which, however, can be significantly alleviated resorting to code parallelization.