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Research Papers

A Sequential Sampling Algorithm for Multistage Static Coverage Problems

[+] Author and Article Information
Binbin Zhang

MAD LAB,
Mechanical and Aerospace Engineering,
University at Buffalo, SUNY,
Buffalo, NY 14260
e-mail: bzhang25@buffalo.edu

Jida Huang

Industrial and Systems Engineering,
University at Buffalo, SUNY,
Buffalo, NY 14260
e-mail: jidahuan@buffalo.edu

Rahul Rai

MAD LAB,
Mechanical and Aerospace Engineering,
University at Buffalo, SUNY,
Buffalo, NY 14260
e-mail: rahulrai@buffalo.edu

Hemanth Manjunatha

Mechanical and Aerospace Engineering,
University at Buffalo, SUNY,
Buffalo, NY 14260
e-mail: hemanthm@buffalo.edu

1Corresponding author.

Manuscript received November 3, 2017; final manuscript received April 4, 2018; published online April 30, 2018. Assoc. Editor: Conrad Tucker.

J. Comput. Inf. Sci. Eng 18(2), 021016 (Apr 30, 2018) (10 pages) Paper No: JCISE-17-1257; doi: 10.1115/1.4039901 History: Received November 03, 2017; Revised April 04, 2018

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.

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Figures

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Fig. 1

Inputs and output of the MSCP: (a) inputs: RIAM function and an initial set of agents and (b) output: optimized locations of the new agents

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Fig. 2

δ1 optimization function: (a) Delaunay triangulation and Voronoi diagram of vertices (solid circles), (b) the constructed Delaunay triangulation surface f̂DT(·), (c) the given RIAM function f(⋅), (d) uniform sampling grid, and (e) the computed error surface that leads to calculation of δ1

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Fig. 3

Overview of the solution methodology for solving MSCPs

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Fig. 4

Pattern search-based approach for solving MSCP

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Fig. 5

Two-peak Gaussian function

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Fig. 6

The placement of agents arrived byusing the PS algorithm and (a) L2 function, (b) SMED function, and (c) combined objective function with l = 25, m = 25, and T = 1

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Fig. 7

Surface plot for complex non-Gaussian distribution function

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Fig. 8

Contour plot of Eq. (9) with optimal samples allocated in three stages. The black triangles represent initial agents l = 25; blue circles represent first-stage agents mT=1 = 10; green squares represent second-stage agents mT=2 = 10; red stars represent third-stage agents mT=3 = 10. (See color figure online.)

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Fig. 9

The variation of objective function with respect to iterations for Eq. (9). The solid line, dash line, and dot line represent first, second, and third stages, respectively.

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Fig. 10

The variation of objective function with respect to iterations higher dimensional problem

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Fig. 11

U-wind data contour plot and its fitted Kriging model: (a) contour plot of U-wind data [35] and (b) Kriging metamodel of wind data

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Fig. 12

Contour plot of U-wind data with optimal samples allocated in three stages. The black triangles represent initial agents l = 25; blue circles represent first-stage agents mT=1 = 10; green squares represent second-stage agents mT=2 = 10; red stars represent third-stage agents mT=3 = 10. (See color figure online.)

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Fig. 13

The variation of objective function with respect to iterations for windmill placement problem. The solid line, dash line, and dot line represent first, second, and third stages, respectively.

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