Safety instrumented systems must be designed, built and operated to meet tolerable risk level as required regulatory agencies. This requirement is closely related to their probabilistic performance measures which are either their average probability of dangerous failure on demand (PFDavg) or their average frequency of failure (PFH: Probability of Failure per Hour). The object of this work is the SIS performances evaluation taking into account the uncertainties related to the different parameters that come into play: failure rate (λ), common cause failure proportion (β), diagnostic coverage (DC), etc. This leads to an accurate and therefore safe assessment of the safety integrity level (SIL) inherent to safety functions performed by such systems. This aim is in keeping with the requirement of the IEC 61508 standard with respect to handling uncertainty. In this paper we first explain in detail the IEC 61508 approach for handling uncertainty. Afterwards, we propose an approach that combines (i) Monte Carlo analysis (MCA) and (ii) fuzzy sets. Indeed, the first method is appropriate when representative statistical data are available (using pdf of the relating parameters), while the latter applies in the case characterized by vague and subjective information (using membership function). The proposed approach is fully supported with a suitable computer code developed under the MATLAB environment.

Safety instrumented systems must be designed, built and operated to meet tolerable risk level as required regulatory agencies. This requirement is closely related to their probabilistic performance measures which are either their average probability of dangerous failure on demand (PFDavg) or their average frequency of failure (PFH: Probability of Failure per Hour). The object of this work is the SIS performances evaluation taking into account the uncertainties related to the different parameters that come into play: failure rate (λ), common cause failure proportion (β), diagnostic coverage (DC), etc. This leads to an accurate and therefore safe assessment of the safety integrity level (SIL) inherent to safety functions performed by such systems. This aim is in keeping with the requirement of the IEC 61508 standard with respect to handling uncertainty. In this paper we first explain in detail the IEC 61508 approach for handling uncertainty. Afterwards, we propose an approach that combines (i) Monte Carlo analysis (MCA) and (ii) fuzzy sets. Indeed, the first method is appropriate when representative statistical data are available (using pdf of the relating parameters), while the latter applies in the case characterized by vague and subjective information (using membership function). The proposed approach is fully supported with a suitable computer code developed under the MATLAB environment.

We consider a system of transmission of the wave equation with Neumann feedback control that contains a delay term and acts on the exterior boundary. First, we prove under some assumptions that the closed-loop system generates a C 0 -semigroup of contractions on an appropriate Hilbert space. Then, under further assumptions, we show that the closed-loop system is exponentially stable. To establish this result, we introduce a suitable energy function and use multiplier method together with an estimate taken from Lasiecka & Triggiani (1992, Appl. Math. Optim., 25, 189–244.) (Lemma 7.2) and compactness-uniqueness argument.

We consider a system of transmission of the wave equation with Neumann feedback control that contains a delay term and acts on the exterior boundary. First, we prove under some assumptions that the closed-loop system generates a C 0 -semigroup of contractions on an appropriate Hilbert space. Then, under further assumptions, we show that the closed-loop system is exponentially stable. To establish this result, we introduce a suitable energy function and use multiplier method together with an estimate taken from Lasiecka & Triggiani (1992, Appl. Math. Optim., 25, 189–244.) (Lemma 7.2) and compactness-uniqueness argument.

In radiotherapy using 18-fluorodeoxyglucose positron emission tomography (18F-FDG-PET), the accurate delineation of the biological tumour volume (BTV) is a crucial step. In this study, the authors suggest a new approach to segment the BTV in 18F-FDG-PET images. The technique is based on the k-means clustering algorithm incorporating automatic optimal cluster number estimation, using intrinsic positron emission tomography image information. Clinical dataset of seven patients have a laryngeal tumour with the actual BTV defined by histology serves as a reference, were included in this study for the evaluation of results. Promising results obtained by the proposed approach with a mean error equal to (0.7%) compared with other existing methods in clinical routine, including fuzzy c-means with (35.58%), gradient-based method with (19.14%) and threshold-based methods.

In radiotherapy using 18-fluorodeoxyglucose positron emission tomography (18F-FDG-PET), the accurate delineation of the biological tumour volume (BTV) is a crucial step. In this study, the authors suggest a new approach to segment the BTV in 18F-FDG-PET images. The technique is based on the k-means clustering algorithm incorporating automatic optimal cluster number estimation, using intrinsic positron emission tomography image information. Clinical dataset of seven patients have a laryngeal tumour with the actual BTV defined by histology serves as a reference, were included in this study for the evaluation of results. Promising results obtained by the proposed approach with a mean error equal to (0.7%) compared with other existing methods in clinical routine, including fuzzy c-means with (35.58%), gradient-based method with (19.14%) and threshold-based methods.

In radiotherapy using 18-fluorodeoxyglucose positron emission tomography (18F-FDG-PET), the accurate delineation of the biological tumour volume (BTV) is a crucial step. In this study, the authors suggest a new approach to segment the BTV in 18F-FDG-PET images. The technique is based on the k-means clustering algorithm incorporating automatic optimal cluster number estimation, using intrinsic positron emission tomography image information. Clinical dataset of seven patients have a laryngeal tumour with the actual BTV defined by histology serves as a reference, were included in this study for the evaluation of results. Promising results obtained by the proposed approach with a mean error equal to (0.7%) compared with other existing methods in clinical routine, including fuzzy c-means with (35.58%), gradient-based method with (19.14%) and threshold-based methods.

In radiotherapy using 18-fluorodeoxyglucose positron emission tomography (18F-FDG-PET), the accurate delineation of the biological tumour volume (BTV) is a crucial step. In this study, the authors suggest a new approach to segment the BTV in 18F-FDG-PET images. The technique is based on the k-means clustering algorithm incorporating automatic optimal cluster number estimation, using intrinsic positron emission tomography image information. Clinical dataset of seven patients have a laryngeal tumour with the actual BTV defined by histology serves as a reference, were included in this study for the evaluation of results. Promising results obtained by the proposed approach with a mean error equal to (0.7%) compared with other existing methods in clinical routine, including fuzzy c-means with (35.58%), gradient-based method with (19.14%) and threshold-based methods.