American Society of Civil Engineers

Formulating and Analyzing Multi-Stage Sensor Placement Problems

by Jean-Paul Watson, (Sandia National Laboratories, Albuquerque, New Mexico. E-mail:, William E. Hart, (Sandia National Laboratories, Albuquerque, New Mexico. E-mail:, David L. Woodruff, (University of California Davis, Graduate School of Management, Davis, California. E-mail:, and Regan Murray, (Environmental Protection Agency, Cincinnati, Ohio. E-mail:
Section: Water Quality, pp. 347-354, (doi:

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Document type: Conference Proceeding Paper
Part of: Water Distribution Systems Analysis 2010
Abstract: The optimization of sensor placements is a key aspect of the design of contaminant warning systems for automatically detecting contaminants in water distribution systems. Although researchers have generally assumed that all sensors are placed at the same time, in practice sensor networks will likely grow and evolve over time. For example, limitations for a water utility’s budget may dictate an staged, incremental deployment of sensors over many years. We describe optimization formulations of multi-stage sensor placement problems. The objective of these formulations includes an explicit trade-off between the value of the initially deployed and final sensor networks. This trade-off motivates the deployment of sensors in initial stages of the deployment schedule, even though these choices typically lead to a solution that is suboptimal when compared to placing all sensors at once. These multi-stage sensor placement problems can be represented as mixed-integer programs, and we illustrate the impact of this trade-off using standard commercial solvers. We also describe a multi-stage formulation that models budget uncertainty, expressed as a tree of potential budget scenarios through time. Budget uncertainty is used to assess and hedge against risks due to a potentially incomplete deployment of a planned sensor network. This formulation is a multi-stage stochastic mixed-integer program, which are notoriously difficult to solve. We apply standard commercial solvers to small-scale test problems, enabling us to effectively analyze multi-stage sensor placement problems subject to budget uncertainties, and assess the impact of accounting for such uncertainty relative to a deterministic multi-stage model.

ASCE Subject Headings:
Probe instruments
Water pollution
Water distribution systems