Istituto per le Applicazioni
della Matematica e dell'Informatica

via A. M. Ampère 56 - 20131 Milano (Italy)

Giornata sull'affidabilità

Martedì 25 Maggio 1999, presso la sede del C.N.R. (via Ampère 56, Milano), aula Convegni, si svolgerà una giornata di studio sul tema dell'affidabilità.


10.00 - 10.45 Impiego di algoritmi genetici per la progettazione ottimale di impianto
Dipartimento di Ingegneria Nucleare, Politecnico di Milano
11.00 - 11.45 Analisi affidabilistica di sistema con metodo Monte Carlo in presenza
di invecchiamento, manutenzioni e obsolescenza tecnologica
Enrico ZIO,
Dipartimento di Ingegneria Nucleare, Politecnico di Milano
12.00 - 12. 45 Analisi di sensitività: un ingrediente essenziale
per la costruzione di modelli matematici e computazionali
EU Joint Research Center, Ispra
14.00 - 14.45 Analisi bayesiana dell'affidabilità di sistemi riparabili
Fabrizio RUGGERI,
CNR-IAMI, Milano
15.00 - 15.45 Un approccio bayesiano robusto per la valutazione della probabilità
di rottura di tubazioni per la distribuzione del gas in ambito metropolitano
Franco CARON, Enrico CAGNO e Mauro MANCINI,
Dipartimento di Meccanica, Politecnico di Milano
16.00 - 16.45 Un modello per la descrizione dei guasti in sistemi riparabili complessi
CNR-IAMI, Milano


Impiego di algoritmi genetici per la progettazione ottimale di impianto

When designing a system, several choices must be made concerning the type of components to be used and their assembly configuration. The choice is driven by the interaction of reliability/availability objectives with the economic costs associated to the design implementation, system construction and future operation. Optimization approaches to determine optimal solutions to design problems have included dynamic programming, integer programming, mixed integer and nonlinear programming, and heuristics. A review of the work in this field can be found in Ref. 1 and 2. In particular, the redundancy allocation problem has been studied extensively [Ref. 3-6] but generally the configurations considered did not include k-out-of-n: G redundancies, whereas this is the case in many practical systems.
In this work we investigate an approach based on the use of genetic algorithms, to determining the optimal system configuration, where the choices can include also k-out-of-n: G schemes. Genetic algorithms are computational tools founded on a direct analogy with the physical evolution of species. An initial population of potential solutions, in the form of coded bit strings, is created and allowed to evolve over successive generations, through mating and mutation. These operations allow the algorithm to explore the state space in an efficient manner. Genetic algorithms have been used to solve several engineering problems and are particularly effective for combinatorial optimization problems with large, complex search spaces. Within the reliability field, however, there have been very few examples of their use. In our work, the objective function used to measure the fitness of a proposed solution is the net profit of system operation for a given mission time. The net profit is obtained by subtracting from the service revenue all the costs associated to the system implementation and operation, i.e. component acquisition and repair costs, system downtime costs, accident costs to restore external environmental conditions and refund from losses in case of an accident. The objective function so designed accounts implicitly for any availability and reliability constraints through the system downtime and accident costs, respectively. Mathematically, then, the problem becomes a search in the system configuration space of that configuration which maximises the objective function.
The optimization algorithm described above is applied to a simple system, for validation purposes. The system is composed of three macro-components A, B and C in series. For each macro-component a choice is to be made among four different redundant configurations, including k-out-of-n: G schemes. The validation is performed under the assumptions that the components cannot be repaired and that failure of macro-component C leads to an accident with external consequences. This last feature extends the application to hazardous production plants such as those in the nuclear or chemical fields. Under these assumptions, the objective function can be computed analytically and the configuration which maximises it can be found by inspection: the results thereby obtained are compared to those obtained by the genetic algorithm and confirm the good performance of the methodology implemented.


1. F. A. Tillman, C. L. Hwang, W. Kuo, Optimization of System Reliability, 1980; Marcel Dekker.

2. F. A. Tillman, C. L. Hwang, W. Kuo, Optimization Techniques for System Reliability with Redundancy: A Review, IEEE Trans. Reliability, vol. R-26, 1977 Aug., pp. 148-155.

3. M.S. Chern, On the Computational Complexity of Reliability Redundancy Allocation in a Series System, Operations Research Letters, Vol. 11, 1992 Jun., pp. 309 - 315.

4. R.E. Bellman, E. Dreyfus, Dynamic Programming and Reliability of Multicomponent Devices, Operations Research, Vol. 6, 1958 Mar - Apr, pp. 200 - 206.

5. D.E. Fyffe, W.W. Hines, N.K. Lee, System Reliability Allocation and a Computational Algorithm, IEEE Trans. Reliability, Vol. R-17, 1968 Jun., pp. 64 - 69.

6. Y. Nakagawa, S. Miyazaki, Surrogate Constraints Algorithm for Reliability Optimization Problems with Two Constraints, IEEE Trans. Reliability, Vol R-30, 1981 Jun, pp. 175 - 180.

Enrico ZIO
Analisi affidabilistica di sistema con metodo Monte Carlo in presenza di invecchiamento, manutenzioni e obsolescenza tecnologica

This work stems from the consideration that a realistic analysis of the safety and reliability of a system should include the influential aspects of aging, maintenance and technological obsolescence. When doing so, however, one should also take into account the economical constraints relating to the plant operation and management which are typically in contrast with the above issues of safety. Modeling all the above safety/economical issues is quite a difficult, if not impossible, task from an analytical viewpoint. Instead, the Monte Carlo approach seems well suited for the task, thanks to its flexibility which allows one to explicitly include, in quite a straightforward way, many of the phenomenological aspects of the problem. Here we present a Monte Carlo simulation scheme for modeling the unavailability of a system, taking into account aging, maintenance and technological obsolescence.

Analisi di sensitività: un ingrediente essenziale per la costruzione di modelli matematici e computazionali

The modelling process following Rosen
Sensitivity Analysis (SA) as tool versus SA as an end (Rabitz versus Oreskes)
The dry-cleaner bill model
About partitioning uncertainty among different sources
Setting and strategies
- Fixing unimportant factors
- Testing the relevance of models
- To falsify - corroborate a model or to select among different models
- As a pre-calibration tool
- To optimise R&D
- To partition epistemic from stochastic uncertainties
- For quality assurance and model reliability
Links of SA with Monte Carlo filtering and "Generalised SA" (Young)
Five properties for an ideal global SA method
- Model independence
- Influence of scale and shape
- Multidimensional averaging
- Capture of interaction effects
- Capacity to treat "sets" of factors as one
The new quasi-ideal methods
- Sobol'
- Extended FAST
- Level E
- ARIMA models

Fabrizio RUGGERI
Analisi bayesiana dell'affidabilità di sistemi riparabili

We will review current research on the Bayesian analysis of the reliability of repairable systems, i.e. systems that can be repaired after a failure (possibly of a small component of it), keeping the same reliability level as before the failure. Such a condition is often called `bad-as-old', even if some authors call it `same-as-old'. Repair time is considered negligible, at least with respect to the operating time of the system.

Data from repairable systems can be available in different forms. We could have a unique system, recording its failures up to a given time T (`time-truncated' data) or until a given number of failures occur (`failure-truncated' data). Besides, we could consider k identical systems for which just (random) operating time and number of failures in it are available. Covariates can be introduced either in the parameters or in their distributions when a hierarchical model is considered. In all these cases, failures of repairable systems are often described by means of a non homogeneous Poisson process, namely the Power Law Process.

Franco CARON, Enrico CAGNO e Mauro MANCINI
Un approccio bayesiano robusto per la valutazione della probabilità di rottura di tubazioni per la distribuzione del gas in ambito metropolitano

We propose a robust Bayesian methodology to assess the probability of failure in different kinds of low-pressure cast-iron pipeline used in metropolitan gas distribution networks. In this respect, after the identification of the factors leading to failure, the main problem is the historical data on failures, which is generally limited and incomplete, and often collected for other purposes. Consequently, effective evaluation of the probability of failure must be based on the integration of historical data and knowledge of company experts. A real world case study is presented in which the company expertise has been elicited by an ad hoc questionnaire and combined with the historical data by means of Bayesian inference. The robustness of the methodology has also been tested.

Un modello per la descrizione dei guasti in sistemi riparabili complessi

This work originates from a consultancy project in a non-standard context: modelling the failure process in a gas distribution network whose size is increasing with time. The available data consist in the failure times, the total length of the network at the beginning of each year since the network started operating, and the laying dates of the failed pipes. The network is divided into subnetworks according to the laying date.
Each failure is then assigned to a subnetwork according to the laying date of the failed pipe, and the failure process in each subnetwork is modelled as a power law process independently of the other subnetworks. The failure process of the whole network is obtained by superimposing the power law processes of all the subnetworks.
The failure-to-subnetwork assignment is made because the network is believed to be a system with increasing failure rate, so the laying date of a pipe is relevant.
The problem also requires the treatment of missing information, that is, some of the installation dates, so that all the failures can then be assigned.
Parameter estimation and the treatment of missing data are performed via standard Bayesian methods, where the necessary computation is carried out with MCMC methods.
Two alternative models are proposed: one where all the information on the laying dates is ignored, and another one where all the available information is incorporated in the model. The two models are proved to coincide when all the dates are missing and are uniformly distributed.