Designing cost-efficient inspection schemes for stochastic streamflow environment using an effective Hamiltonian approach

Designing inspection frequency to efficiently track stochastic dynamics is a fundamental engineering problem. Especially, tracking environmental variables like water quantity and quality is of vital importance for sustainable and sound development. However, theoretical understanding of methodologies to design cost-efficient inspection schemes of environmental dynamics is still limited. To tackle this issue, a new Poison inspection problem focusing on coupled streamflow and water quality dynamics in rivers was proposed as a model problem of designing inspection schemes in industries. The coupled dynamics are represented by Levy-driven stochastic differential equations. A long-run performance index, an effective Hamiltonian, containing inspection cost and penalization of the information loss between successive inspections is then formulated. The design variable of the proposed model is the inspection frequency. Our optimization problem has two levels where the effective Hamiltonian is firstly obtained in a closed-form from a Hamilton − Jacobi − Bellman equation. The effective Hamiltonian is then minimized concerning the inspection frequency. An application of the proposed model to designing cost-efficient inspection schemes of dissolved silica concentration, a key index in inland fishery industries, of a river in Japan, is also discussed with unique real data.