(248f) Optimal Placement of Imperfect Detectors in the Design of Mitigation Systems: A Non-Uniform Unavailability P-Median Formulation

Mitigation systems are designed to act as the last layer of protection against health, personnel, and environmental hazards. Several of these mitigation systems rely on properly placed detectors to efficiently acknowledge the hazards before issuing executive actions.  One such example is the placement of gas detectors within chemical process facilities to provide rapid detection of flammable or toxic gas releases. Most strategies in use for design of mitigation systems like this are based on qualitative or semi-quantitative approaches rather than in the rigorous quantitative consideration of the hazardous scenarios. To address this gap, while considering the inherent uncertainty associated with mitigation systems, the use of a stochastic programming formulation (SP) equivalent to the P-median Problem (PMP) was proposed and validated in the mitigation of contamination scenarios within water distribution systems.  This formulation has also been validated for optimal design of other types of mitigation systems. The use of an extended version of formulation SP resulted in an order of magnitude improvement in the performance of gas detection systems when compared to some of the prescriptive detector placement approaches used in the industry. These validation and comparison studies were performed using real sets of CFD generated data provided by GexCon. The data corresponds to FLACS dispersion simulations including the full geometric features of an offshore facility and capturing the uncertainty in the leak characteristics.

Previous work by the authors extended formulation SP into formulation SPU to allow the consideration of imperfect detection (detector unavailability). Unavailability represents the probability of a false negative, i.e., the probability that the detector will not be able to perform its intended function when required. Detector unavailability impacts the system performance, and explicit consideration of detector unavailability affects the optimal placement of detectors. Moreover, neglecting detector unavailability when optimizing the layout will result in a significant deterioration of the mitigation system effectiveness in realistic situations. Formulation SPU assumes that all the detectors have a uniform unavailability, that is, an identical unavailability is assumed for all detectors. In this work, we present formulation SPq, an extended version of formulation SPU that relaxes this assumption allowing for non-uniform detector unavailabilities. This leads to a mixed-integer nonlinear programming formulation due to the need to model products of non-uniform detector unavailabilities. However, formulation SPq explicitly considers individual detection levels, allowing the modeler to specify the number of detection levels, dramatically reduce the computational complexity of the problem. Furthermore, the effect of truncating the number of detection levels for the real gas detector mitigation system problem was analyzed. Based on this analysis, a quadratic formulation that provides a good compromise between accuracy and complexity is presented. Three solution strategies were proposed for this quadratic problem: SPq-Q, SPq-L1, and SPq-L2. The quadratic formulation performance was tested on the same real sets of CFD generated data. From a solution quality perspective, the proposed quadratic formulation outperforms previously proposed approximate solution strategies in the literature.