A Sequential Sampling Algorithm for Multistage Static Coverage Problems

In many system-engineering problems, such as surveillance, environmental monitoring, and cooperative task performance, it is critical to allocate limited resources within a restricted area optimally. Static coverage problem (SCP) is an important class of the resource allocation problem. SCP focuses on covering an area of interest so that the activities in that area can be detected with high probabilities. In many practical settings, primarily due to financial constraints, a system designer has to allocate resources in multiple stages. In each stage, the system designer can assign a fixed number of resources, i.e., agents. In the multistage formulation, agent locations for the next stage are dependent on previous-stage agent locations. Such multistage static coverage problems are nontrivial to solve. In this paper, we propose an efficient sequential sampling algorithm to solve the multistage static coverage problem (MSCP) in the presence of resource intensity allocation maps (RIAMs) distribution functions that abstract the event that we want to detect/monitor in a given area. The agent's location in the successive stage is determined by formulating it as an optimization problem. Three different objective functions have been developed and proposed in this paper: (1) L2 difference, (2) sequential minimum energy design (SMED), and (3) the weighted L2 and SMED. Pattern search (PS), an efficient heuristic algorithm has been used as optimization algorithm to arrive at the solutions for the formulated optimization problems. The developed approach has been tested on two- and higher dimensional functions. The results analyzing real-life applications of windmill placement inside a wind farm in multiple stages are also presented.